:: ENS_1 semantic presentation

K170() is Element of bool K166()
K166() is set
bool K166() is non empty set
{} is Relation-like non-empty empty-yielding Function-like one-to-one constant functional empty set
1 is non empty set
{{},1} is non empty set
K111() is set
bool K111() is non empty set
bool K170() is non empty set
[:1,1:] is Relation-like non empty set
bool [:1,1:] is non empty set
[:[:1,1:],1:] is Relation-like non empty set
bool [:[:1,1:],1:] is non empty set
V is non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
V is non empty set
(V) is set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
the Element of V is Element of V
id the Element of V is Relation-like the Element of V -defined the Element of V -valued Function-like one-to-one total quasi_total Element of bool [: the Element of V, the Element of V:]
[: the Element of V, the Element of V:] is Relation-like set
bool [: the Element of V, the Element of V:] is non empty set
Funcs ( the Element of V, the Element of V) is functional non empty set
A is set
b is non empty set
d is set
g9 is Element of V
f is Element of V
Funcs (g9,f) is functional set
V is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
C is set
b is set
A is Element of V
d is Element of V
Funcs (A,d) is functional set
[:A,d:] is Relation-like set
bool [:A,d:] is non empty set
A is Element of V
b is Element of V
[:b,A:] is Relation-like set
bool [:b,A:] is non empty set
Funcs (b,A) is functional set
V is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
C is Element of V
a is Element of V
Funcs (C,a) is functional set
b is set
[:C,a:] is Relation-like set
bool [:C,a:] is non empty set
V is non empty set
bool V is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
C is non empty Element of bool V
(C) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of C : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of C : verum } is set
a is set
A is Element of C
b is Element of C
[:b,A:] is Relation-like set
bool [:b,A:] is non empty set
V is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
V is non empty set
(V) is set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
the Element of V is Element of V
id the Element of V is Relation-like the Element of V -defined the Element of V -valued Function-like one-to-one total quasi_total Element of bool [: the Element of V, the Element of V:]
[: the Element of V, the Element of V:] is Relation-like set
bool [: the Element of V, the Element of V:] is non empty set
[:[:V,V:],(bool [: the Element of V, the Element of V:]):] is Relation-like non empty set
[ the Element of V, the Element of V] is V15() Element of [:V,V:]
{ the Element of V, the Element of V} is non empty set
{ the Element of V} is non empty set
{{ the Element of V, the Element of V},{ the Element of V}} is non empty set
[[ the Element of V, the Element of V],(id the Element of V)] is V15() Element of [:[:V,V:],(bool [: the Element of V, the Element of V:]):]
{[ the Element of V, the Element of V],(id the Element of V)} is non empty set
{[ the Element of V, the Element of V]} is Relation-like Function-like non empty set
{{[ the Element of V, the Element of V],(id the Element of V)},{[ the Element of V, the Element of V]}} is non empty set
A is V15() Element of [:[:V,V:],(bool [: the Element of V, the Element of V:]):]
b is V15() Element of [:[:V,V:],(bool [: the Element of V, the Element of V:]):]
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
C is Element of (V)
a is Element of V
b is Element of V
[a,b] is V15() Element of [:V,V:]
{a,b} is non empty set
{a} is non empty set
{{a,b},{a}} is non empty set
A is Relation-like Function-like Element of (V)
[[a,b],A] is V15() Element of [:[:V,V:],(V):]
[:[:V,V:],(V):] is Relation-like non empty set
{[a,b],A} is non empty set
{[a,b]} is Relation-like Function-like non empty set
{{[a,b],A},{[a,b]}} is non empty set
[:a,b:] is Relation-like set
bool [:a,b:] is non empty set
V is non empty set
[:V,V:] is Relation-like non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
C is Element of V
a is Element of V
[:C,a:] is Relation-like set
bool [:C,a:] is non empty set
[C,a] is V15() Element of [:V,V:]
{C,a} is non empty set
{C} is non empty set
{{C,a},{C}} is non empty set
b is Relation-like C -defined a -valued Function-like quasi_total Element of bool [:C,a:]
[[C,a],b] is V15() Element of [:[:V,V:],(bool [:C,a:]):]
[:[:V,V:],(bool [:C,a:]):] is Relation-like non empty set
{[C,a],b} is non empty set
{[C,a]} is Relation-like Function-like non empty set
{{[C,a],b},{[C,a]}} is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
[:[:V,V:],(V):] is Relation-like non empty set
C is set
a is Element of V
b is Element of V
[a,b] is V15() Element of [:V,V:]
{a,b} is non empty set
{a} is non empty set
{{a,b},{a}} is non empty set
A is Relation-like Function-like Element of (V)
[[a,b],A] is V15() Element of [:[:V,V:],(V):]
{[a,b],A} is non empty set
{[a,b]} is Relation-like Function-like non empty set
{{[a,b],A},{[a,b]}} is non empty set
[:a,b:] is Relation-like set
bool [:a,b:] is non empty set
V is non empty set
bool V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
C is non empty Element of bool V
(C) is non empty set
(C) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of C : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of C : verum } is set
[:C,C:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of C, b₃ is Relation-like Function-like Element of (C) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
a is set
b is Element of C
A is Element of C
[b,A] is V15() Element of [:C,C:]
{b,A} is non empty set
{b} is non empty set
{{b,A},{b}} is non empty set
d is Relation-like Function-like Element of (C)
[[b,A],d] is V15() Element of [:[:C,C:],(C):]
[:[:C,C:],(C):] is Relation-like non empty set
{[b,A],d} is non empty set
{[b,A]} is Relation-like Function-like non empty set
{{[b,A],d},{[b,A]}} is non empty set
[:b,A:] is Relation-like set
bool [:b,A:] is non empty set
V is set
a is set
[V,a] is V15() set
{V,a} is non empty set
{V} is non empty set
{{V,a},{V}} is non empty set
A is set
[[V,a],A] is V15() set
{[V,a],A} is non empty set
{[V,a]} is Relation-like Function-like non empty set
{{[V,a],A},{[V,a]}} is non empty set
C is set
b is set
[C,b] is V15() set
{C,b} is non empty set
{C} is non empty set
{{C,b},{C}} is non empty set
d is set
[[C,b],d] is V15() set
{[C,b],d} is non empty set
{[C,b]} is Relation-like Function-like non empty set
{{[C,b],d},{[C,b]}} is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
C is Element of (V)
C `2 is set
b is Element of V
A is Element of V
[b,A] is V15() Element of [:V,V:]
{b,A} is non empty set
{b} is non empty set
{{b,A},{b}} is non empty set
a is Relation-like Function-like Element of (V)
[[b,A],a] is V15() Element of [:[:V,V:],(V):]
[:[:V,V:],(V):] is Relation-like non empty set
{[b,A],a} is non empty set
{[b,A]} is Relation-like Function-like non empty set
{{[b,A],a},{[b,A]}} is non empty set
[:b,A:] is Relation-like set
bool [:b,A:] is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
C is Element of (V)
C `1 is set
(C `1) `1 is set
b is Element of V
A is Element of V
[b,A] is V15() Element of [:V,V:]
{b,A} is non empty set
{b} is non empty set
{{b,A},{b}} is non empty set
a is Relation-like Function-like Element of (V)
[[b,A],a] is V15() Element of [:[:V,V:],(V):]
[:[:V,V:],(V):] is Relation-like non empty set
{[b,A],a} is non empty set
{[b,A]} is Relation-like Function-like non empty set
{{[b,A],a},{[b,A]}} is non empty set
[:b,A:] is Relation-like set
bool [:b,A:] is non empty set
(C `1) `2 is set
b is Element of V
A is Element of V
[b,A] is V15() Element of [:V,V:]
{b,A} is non empty set
{b} is non empty set
{{b,A},{b}} is non empty set
a is Relation-like Function-like Element of (V)
[[b,A],a] is V15() Element of [:[:V,V:],(V):]
[:[:V,V:],(V):] is Relation-like non empty set
{[b,A],a} is non empty set
{[b,A]} is Relation-like Function-like non empty set
{{[b,A],a},{[b,A]}} is non empty set
[:b,A:] is Relation-like set
bool [:b,A:] is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
C is Element of (V)
(V,C) is Element of V
C `1 is set
(C `1) `1 is set
(V,C) is Element of V
(C `1) `2 is set
[(V,C),(V,C)] is V15() Element of [:V,V:]
{(V,C),(V,C)} is non empty set
{(V,C)} is non empty set
{{(V,C),(V,C)},{(V,C)}} is non empty set
C `2 is Relation-like Function-like set
[[(V,C),(V,C)],(C `2)] is V15() set
{[(V,C),(V,C)],(C `2)} is non empty set
{[(V,C),(V,C)]} is Relation-like Function-like non empty set
{{[(V,C),(V,C)],(C `2)},{[(V,C),(V,C)]}} is non empty set
b is Element of V
A is Element of V
[b,A] is V15() Element of [:V,V:]
{b,A} is non empty set
{b} is non empty set
{{b,A},{b}} is non empty set
a is Relation-like Function-like Element of (V)
[[b,A],a] is V15() Element of [:[:V,V:],(V):]
[:[:V,V:],(V):] is Relation-like non empty set
{[b,A],a} is non empty set
{[b,A]} is Relation-like Function-like non empty set
{{[b,A],a},{[b,A]}} is non empty set
[:b,A:] is Relation-like set
bool [:b,A:] is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
C is Element of (V)
C `2 is Relation-like Function-like set
(V,C) is Element of V
C `1 is set
(C `1) `1 is set
(V,C) is Element of V
(C `1) `2 is set
a is Element of (V)
a `2 is Relation-like Function-like set
(V,a) is Element of V
a `1 is set
(a `1) `1 is set
(V,a) is Element of V
(a `1) `2 is set
[(V,C),(V,C)] is V15() Element of [:V,V:]
{(V,C),(V,C)} is non empty set
{(V,C)} is non empty set
{{(V,C),(V,C)},{(V,C)}} is non empty set
[[(V,C),(V,C)],(C `2)] is V15() set
{[(V,C),(V,C)],(C `2)} is non empty set
{[(V,C),(V,C)]} is Relation-like Function-like non empty set
{{[(V,C),(V,C)],(C `2)},{[(V,C),(V,C)]}} is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
C is Element of (V)
(V,C) is Element of V
C `1 is set
(C `1) `2 is set
(V,C) is Element of V
(C `1) `1 is set
C `2 is Relation-like Function-like set
[:(V,C),(V,C):] is Relation-like set
bool [:(V,C),(V,C):] is non empty set
b is Element of V
A is Element of V
[b,A] is V15() Element of [:V,V:]
{b,A} is non empty set
{b} is non empty set
{{b,A},{b}} is non empty set
a is Relation-like Function-like Element of (V)
[[b,A],a] is V15() Element of [:[:V,V:],(V):]
[:[:V,V:],(V):] is Relation-like non empty set
{[b,A],a} is non empty set
{[b,A]} is Relation-like Function-like non empty set
{{[b,A],a},{[b,A]}} is non empty set
[:b,A:] is Relation-like set
bool [:b,A:] is non empty set
[(V,C),(V,C)] is V15() Element of [:V,V:]
{(V,C),(V,C)} is non empty set
{(V,C)} is non empty set
{{(V,C),(V,C)},{(V,C)}} is non empty set
[[(V,C),(V,C)],(C `2)] is V15() set
{[(V,C),(V,C)],(C `2)} is non empty set
{[(V,C),(V,C)]} is Relation-like Function-like non empty set
{{[(V,C),(V,C)],(C `2)},{[(V,C),(V,C)]}} is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
C is Element of (V)
C `2 is Relation-like Function-like set
proj1 (C `2) is set
(V,C) is Element of V
C `1 is set
(C `1) `1 is set
proj2 (C `2) is set
(V,C) is Element of V
(C `1) `2 is set
[:(V,C),(V,C):] is Relation-like set
bool [:(V,C),(V,C):] is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
C is Relation-like Function-like set
a is set
b is set
[a,b] is V15() set
{a,b} is non empty set
{a} is non empty set
{{a,b},{a}} is non empty set
[[a,b],C] is V15() set
{[a,b],C} is non empty set
{[a,b]} is Relation-like Function-like non empty set
{{[a,b],C},{[a,b]}} is non empty set
[:a,b:] is Relation-like set
bool [:a,b:] is non empty set
d is Element of V
g9 is Element of V
[d,g9] is V15() Element of [:V,V:]
{d,g9} is non empty set
{d} is non empty set
{{d,g9},{d}} is non empty set
A is Relation-like Function-like Element of (V)
[[d,g9],A] is V15() Element of [:[:V,V:],(V):]
[:[:V,V:],(V):] is Relation-like non empty set
{[d,g9],A} is non empty set
{[d,g9]} is Relation-like Function-like non empty set
{{[d,g9],A},{[d,g9]}} is non empty set
[:d,g9:] is Relation-like set
bool [:d,g9:] is non empty set
V is non empty set
[:V,V:] is Relation-like non empty set
C is Element of V
[:C,C:] is Relation-like set
bool [:C,C:] is non empty set
[C,C] is V15() Element of [:V,V:]
{C,C} is non empty set
{C} is non empty set
{{C,C},{C}} is non empty set
id C is Relation-like C -defined C -valued Function-like one-to-one total quasi_total Element of bool [:C,C:]
[[C,C],(id C)] is V15() Element of [:[:V,V:],(bool [:C,C:]):]
[:[:V,V:],(bool [:C,C:]):] is Relation-like non empty set
{[C,C],(id C)} is non empty set
{[C,C]} is Relation-like Function-like non empty set
{{[C,C],(id C)},{[C,C]}} is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
V is non empty set
C is Element of V
(V,C) is Element of (V)
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
[:C,C:] is Relation-like set
bool [:C,C:] is non empty set
[C,C] is V15() Element of [:V,V:]
{C,C} is non empty set
{C} is non empty set
{{C,C},{C}} is non empty set
id C is Relation-like C -defined C -valued Function-like one-to-one total quasi_total Element of bool [:C,C:]
[[C,C],(id C)] is V15() Element of [:[:V,V:],(bool [:C,C:]):]
[:[:V,V:],(bool [:C,C:]):] is Relation-like non empty set
{[C,C],(id C)} is non empty set
{[C,C]} is Relation-like Function-like non empty set
{{[C,C],(id C)},{[C,C]}} is non empty set
(V,C) `2 is Relation-like Function-like set
(V,(V,C)) is Element of V
(V,C) `1 is set
((V,C) `1) `1 is set
(V,(V,C)) is Element of V
((V,C) `1) `2 is set
[(V,(V,C)),(V,(V,C))] is V15() Element of [:V,V:]
{(V,(V,C)),(V,(V,C))} is non empty set
{(V,(V,C))} is non empty set
{{(V,(V,C)),(V,(V,C))},{(V,(V,C))}} is non empty set
[[(V,(V,C)),(V,(V,C))],((V,C) `2)] is V15() set
{[(V,(V,C)),(V,(V,C))],((V,C) `2)} is non empty set
{[(V,(V,C)),(V,(V,C))]} is Relation-like Function-like non empty set
{{[(V,(V,C)),(V,(V,C))],((V,C) `2)},{[(V,(V,C)),(V,(V,C))]}} is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
C is Element of (V)
(V,C) is Element of V
C `1 is set
(C `1) `2 is set
a is Element of (V)
(V,a) is Element of V
a `1 is set
(a `1) `1 is set
(V,C) is Element of V
(C `1) `1 is set
(V,a) is Element of V
(a `1) `2 is set
[(V,C),(V,a)] is V15() Element of [:V,V:]
{(V,C),(V,a)} is non empty set
{(V,C)} is non empty set
{{(V,C),(V,a)},{(V,C)}} is non empty set
C `2 is Relation-like Function-like set
a `2 is Relation-like Function-like set
(C `2) * (a `2) is Relation-like Function-like set
[[(V,C),(V,a)],((C `2) * (a `2))] is V15() set
{[(V,C),(V,a)],((C `2) * (a `2))} is non empty set
{[(V,C),(V,a)]} is Relation-like Function-like non empty set
{{[(V,C),(V,a)],((C `2) * (a `2))},{[(V,C),(V,a)]}} is non empty set
[:(V,C),(V,C):] is Relation-like set
bool [:(V,C),(V,C):] is non empty set
[:(V,a),(V,a):] is Relation-like set
bool [:(V,a),(V,a):] is non empty set
[:(V,C),(V,a):] is Relation-like set
bool [:(V,C),(V,a):] is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
C is Element of (V)
(V,C) is Element of V
C `1 is set
(C `1) `1 is set
C `2 is Relation-like Function-like set
(V,C) is Element of V
(C `1) `2 is set
a is Element of (V)
(V,a) is Element of V
a `1 is set
(a `1) `2 is set
(V,a,C) is Element of (V)
(V,a,C) `2 is Relation-like Function-like set
a `2 is Relation-like Function-like set
(a `2) * (C `2) is Relation-like Function-like set
(V,(V,a,C)) is Element of V
(V,a,C) `1 is set
((V,a,C) `1) `1 is set
(V,a) is Element of V
(a `1) `1 is set
(V,(V,a,C)) is Element of V
((V,a,C) `1) `2 is set
[(V,a),(V,C)] is V15() Element of [:V,V:]
{(V,a),(V,C)} is non empty set
{(V,a)} is non empty set
{{(V,a),(V,C)},{(V,a)}} is non empty set
[[(V,a),(V,C)],((a `2) * (C `2))] is V15() set
{[(V,a),(V,C)],((a `2) * (C `2))} is non empty set
{[(V,a),(V,C)]} is Relation-like Function-like non empty set
{{[(V,a),(V,C)],((a `2) * (C `2))},{[(V,a),(V,C)]}} is non empty set
[(V,(V,a,C)),(V,(V,a,C))] is V15() Element of [:V,V:]
{(V,(V,a,C)),(V,(V,a,C))} is non empty set
{(V,(V,a,C))} is non empty set
{{(V,(V,a,C)),(V,(V,a,C))},{(V,(V,a,C))}} is non empty set
[[(V,(V,a,C)),(V,(V,a,C))],((V,a,C) `2)] is V15() set
{[(V,(V,a,C)),(V,(V,a,C))],((V,a,C) `2)} is non empty set
{[(V,(V,a,C)),(V,(V,a,C))]} is Relation-like Function-like non empty set
{{[(V,(V,a,C)),(V,(V,a,C))],((V,a,C) `2)},{[(V,(V,a,C)),(V,(V,a,C))]}} is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
C is Element of (V)
(V,C) is Element of V
C `1 is set
(C `1) `1 is set
(V,C) is Element of V
(C `1) `2 is set
a is Element of (V)
(V,a) is Element of V
a `1 is set
(a `1) `2 is set
(V,a,C) is Element of (V)
b is Element of (V)
(V,b) is Element of V
b `1 is set
(b `1) `1 is set
(V,(V,a,C),b) is Element of (V)
(V,C,b) is Element of (V)
(V,a,(V,C,b)) is Element of (V)
(V,(V,a,C)) is Element of V
(V,a,C) `1 is set
((V,a,C) `1) `2 is set
(V,a,C) `2 is Relation-like Function-like set
a `2 is Relation-like Function-like set
C `2 is Relation-like Function-like set
(a `2) * (C `2) is Relation-like Function-like set
(V,(V,a,C),b) `2 is Relation-like Function-like set
b `2 is Relation-like Function-like set
((a `2) * (C `2)) * (b `2) is Relation-like Function-like set
(V,(V,C,b)) is Element of V
(V,C,b) `1 is set
((V,C,b) `1) `1 is set
(V,(V,a,(V,C,b))) is Element of V
(V,a,(V,C,b)) `1 is set
((V,a,(V,C,b)) `1) `1 is set
(V,a) is Element of V
(a `1) `1 is set
(V,(V,a,C)) is Element of V
((V,a,C) `1) `1 is set
(V,(V,(V,a,C),b)) is Element of V
(V,(V,a,C),b) `1 is set
((V,(V,a,C),b) `1) `1 is set
(V,(V,C,b)) is Element of V
((V,C,b) `1) `2 is set
(V,b) is Element of V
(b `1) `2 is set
(V,(V,a,(V,C,b))) is Element of V
((V,a,(V,C,b)) `1) `2 is set
(V,C,b) `2 is Relation-like Function-like set
(C `2) * (b `2) is Relation-like Function-like set
(V,a,(V,C,b)) `2 is Relation-like Function-like set
(a `2) * ((C `2) * (b `2)) is Relation-like Function-like set
(V,(V,(V,a,C),b)) is Element of V
((V,(V,a,C),b) `1) `2 is set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
C is Element of (V)
(V,C) is Element of V
C `1 is set
(C `1) `1 is set
(V,(V,C)) is Element of (V)
[:(V,C),(V,C):] is Relation-like set
bool [:(V,C),(V,C):] is non empty set
[(V,C),(V,C)] is V15() Element of [:V,V:]
{(V,C),(V,C)} is non empty set
{(V,C)} is non empty set
{{(V,C),(V,C)},{(V,C)}} is non empty set
id (V,C) is Relation-like (V,C) -defined (V,C) -valued Function-like one-to-one total quasi_total Element of bool [:(V,C),(V,C):]
[[(V,C),(V,C)],(id (V,C))] is V15() Element of [:[:V,V:],(bool [:(V,C),(V,C):]):]
[:[:V,V:],(bool [:(V,C),(V,C):]):] is Relation-like non empty set
{[(V,C),(V,C)],(id (V,C))} is non empty set
{[(V,C),(V,C)]} is Relation-like Function-like non empty set
{{[(V,C),(V,C)],(id (V,C))},{[(V,C),(V,C)]}} is non empty set
(V,(V,(V,C)),C) is Element of (V)
(V,C) is Element of V
(C `1) `2 is set
(V,(V,C)) is Element of (V)
[:(V,C),(V,C):] is Relation-like set
bool [:(V,C),(V,C):] is non empty set
[(V,C),(V,C)] is V15() Element of [:V,V:]
{(V,C),(V,C)} is non empty set
{(V,C)} is non empty set
{{(V,C),(V,C)},{(V,C)}} is non empty set
id (V,C) is Relation-like (V,C) -defined (V,C) -valued Function-like one-to-one total quasi_total Element of bool [:(V,C),(V,C):]
[[(V,C),(V,C)],(id (V,C))] is V15() Element of [:[:V,V:],(bool [:(V,C),(V,C):]):]
[:[:V,V:],(bool [:(V,C),(V,C):]):] is Relation-like non empty set
{[(V,C),(V,C)],(id (V,C))} is non empty set
{[(V,C),(V,C)]} is Relation-like Function-like non empty set
{{[(V,C),(V,C)],(id (V,C))},{[(V,C),(V,C)]}} is non empty set
(V,C,(V,(V,C))) is Element of (V)
(V,(V,C)) `2 is Relation-like Function-like set
(V,(V,(V,C))) is Element of V
(V,(V,C)) `1 is set
((V,(V,C)) `1) `1 is set
C `2 is Relation-like Function-like set
[:(V,C),(V,C):] is Relation-like set
bool [:(V,C),(V,C):] is non empty set
proj2 (C `2) is set
proj1 (C `2) is set
(V,(V,(V,C))) is Element of V
((V,(V,C)) `1) `2 is set
(V,(V,(V,(V,C)),C)) is Element of V
(V,(V,(V,C)),C) `1 is set
((V,(V,(V,C)),C) `1) `2 is set
(V,(V,(V,C)),C) `2 is Relation-like Function-like set
((V,(V,C)) `2) * (C `2) is Relation-like Function-like set
(V,(V,(V,(V,C)),C)) is Element of V
((V,(V,(V,C)),C) `1) `1 is set
(V,(V,C)) `2 is Relation-like Function-like set
(V,(V,(V,C))) is Element of V
(V,(V,C)) `1 is set
((V,(V,C)) `1) `2 is set
(V,(V,(V,C))) is Element of V
((V,(V,C)) `1) `1 is set
(V,(V,C,(V,(V,C)))) is Element of V
(V,C,(V,(V,C))) `1 is set
((V,C,(V,(V,C))) `1) `2 is set
(V,C,(V,(V,C))) `2 is Relation-like Function-like set
(C `2) * ((V,(V,C)) `2) is Relation-like Function-like set
(V,(V,C,(V,(V,C)))) is Element of V
((V,C,(V,(V,C))) `1) `1 is set
V is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
C is Element of V
a is Element of V
[C,a] is V15() Element of [:V,V:]
{C,a} is non empty set
{C} is non empty set
{{C,a},{C}} is non empty set
(V) is non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
{ [[C,a],b<sub>1</sub>] where b<sub>1</sub> is Relation-like Function-like Element of (V) : [[C,a],b<sub>1</sub>] in (V) } is set
V is non empty set
[:V,V:] is Relation-like non empty set
C is Element of V
a is Element of V
[:C,a:] is Relation-like set
bool [:C,a:] is non empty set
[C,a] is V15() Element of [:V,V:]
{C,a} is non empty set
{C} is non empty set
{{C,a},{C}} is non empty set
(V,C,a) is set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
(V) is non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
{ [[C,a],b<sub>1</sub>] where b<sub>1</sub> is Relation-like Function-like Element of (V) : [[C,a],b<sub>1</sub>] in (V) } is set
b is Relation-like C -defined a -valued Function-like quasi_total Element of bool [:C,a:]
[[C,a],b] is V15() Element of [:[:V,V:],(bool [:C,a:]):]
[:[:V,V:],(bool [:C,a:]):] is Relation-like non empty set
{[C,a],b} is non empty set
{[C,a]} is Relation-like Function-like non empty set
{{[C,a],b},{[C,a]}} is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
C is Element of V
a is Element of V
(V,C,a) is set
[C,a] is V15() Element of [:V,V:]
{C,a} is non empty set
{C} is non empty set
{{C,a},{C}} is non empty set
{ [[C,a],b<sub>1</sub>] where b<sub>1</sub> is Relation-like Function-like Element of (V) : [[C,a],b<sub>1</sub>] in (V) } is set
b is Element of (V)
b `2 is Relation-like Function-like set
[[C,a],(b `2)] is V15() set
{[C,a],(b `2)} is non empty set
{[C,a]} is Relation-like Function-like non empty set
{{[C,a],(b `2)},{[C,a]}} is non empty set
(V,b) is Element of V
b `1 is set
(b `1) `1 is set
(V,b) is Element of V
(b `1) `2 is set
[(V,b),(V,b)] is V15() Element of [:V,V:]
{(V,b),(V,b)} is non empty set
{(V,b)} is non empty set
{{(V,b),(V,b)},{(V,b)}} is non empty set
[[(V,b),(V,b)],(b `2)] is V15() set
{[(V,b),(V,b)],(b `2)} is non empty set
{[(V,b),(V,b)]} is Relation-like Function-like non empty set
{{[(V,b),(V,b)],(b `2)},{[(V,b),(V,b)]}} is non empty set
A is Relation-like Function-like Element of (V)
[[C,a],A] is V15() Element of [:[:V,V:],(V):]
[:[:V,V:],(V):] is Relation-like non empty set
{[C,a],A} is non empty set
{{[C,a],A},{[C,a]}} is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
C is Element of V
a is Element of V
(V,C,a) is set
[C,a] is V15() Element of [:V,V:]
{C,a} is non empty set
{C} is non empty set
{{C,a},{C}} is non empty set
{ [[C,a],b<sub>1</sub>] where b<sub>1</sub> is Relation-like Function-like Element of (V) : [[C,a],b<sub>1</sub>] in (V) } is set
b is set
A is Relation-like Function-like Element of (V)
[[C,a],A] is V15() Element of [:[:V,V:],(V):]
[:[:V,V:],(V):] is Relation-like non empty set
{[C,a],A} is non empty set
{[C,a]} is Relation-like Function-like non empty set
{{[C,a],A},{[C,a]}} is non empty set
V is non empty set
C is Element of V
a is Element of V
[C,a] is V15() Element of [:V,V:]
[:V,V:] is Relation-like non empty set
{C,a} is non empty set
{C} is non empty set
{{C,a},{C}} is non empty set
(V,C,a) is set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
(V) is non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
{ [[C,a],b<sub>1</sub>] where b<sub>1</sub> is Relation-like Function-like Element of (V) : [[C,a],b<sub>1</sub>] in (V) } is set
[:C,a:] is Relation-like set
bool [:C,a:] is non empty set
b is Relation-like Function-like set
[[C,a],b] is V15() set
{[C,a],b} is non empty set
{[C,a]} is Relation-like Function-like non empty set
{{[C,a],b},{[C,a]}} is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
{ (V,b<sub>1</sub>,b<sub>2</sub>) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (V,b<sub>1</sub>,b<sub>2</sub>) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
a is set
A is Element of V
d is Element of V
[A,d] is V15() Element of [:V,V:]
{A,d} is non empty set
{A} is non empty set
{{A,d},{A}} is non empty set
b is Relation-like Function-like Element of (V)
[[A,d],b] is V15() Element of [:[:V,V:],(V):]
[:[:V,V:],(V):] is Relation-like non empty set
{[A,d],b} is non empty set
{[A,d]} is Relation-like Function-like non empty set
{{[A,d],b},{[A,d]}} is non empty set
[:A,d:] is Relation-like set
bool [:A,d:] is non empty set
(V,A,d) is set
{ [[A,d],b<sub>1</sub>] where b<sub>1</sub> is Relation-like Function-like Element of (V) : [[A,d],b<sub>1</sub>] in (V) } is set
b is set
A is Element of V
d is Element of V
(V,A,d) is set
[A,d] is V15() Element of [:V,V:]
{A,d} is non empty set
{A} is non empty set
{{A,d},{A}} is non empty set
{ [[A,d],b<sub>1</sub>] where b<sub>1</sub> is Relation-like Function-like Element of (V) : [[A,d],b<sub>1</sub>] in (V) } is set
g9 is Relation-like Function-like Element of (V)
[[A,d],g9] is V15() Element of [:[:V,V:],(V):]
[:[:V,V:],(V):] is Relation-like non empty set
{[A,d],g9} is non empty set
{[A,d]} is Relation-like Function-like non empty set
{{[A,d],g9},{[A,d]}} is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
C is Element of V
a is Element of V
(V,C,a) is set
[C,a] is V15() Element of [:V,V:]
{C,a} is non empty set
{C} is non empty set
{{C,a},{C}} is non empty set
{ [[C,a],b<sub>1</sub>] where b<sub>1</sub> is Relation-like Function-like Element of (V) : [[C,a],b<sub>1</sub>] in (V) } is set
b is Element of (V)
(V,b) is Element of V
b `1 is set
(b `1) `1 is set
(V,b) is Element of V
(b `1) `2 is set
b `2 is Relation-like Function-like set
[:(V,b),(V,b):] is Relation-like set
bool [:(V,b),(V,b):] is non empty set
[[C,a],(b `2)] is V15() set
{[C,a],(b `2)} is non empty set
{[C,a]} is Relation-like Function-like non empty set
{{[C,a],(b `2)},{[C,a]}} is non empty set
[(V,b),(V,b)] is V15() Element of [:V,V:]
{(V,b),(V,b)} is non empty set
{(V,b)} is non empty set
{{(V,b),(V,b)},{(V,b)}} is non empty set
[[(V,b),(V,b)],(b `2)] is V15() set
{[(V,b),(V,b)],(b `2)} is non empty set
{[(V,b),(V,b)]} is Relation-like Function-like non empty set
{{[(V,b),(V,b)],(b `2)},{[(V,b),(V,b)]}} is non empty set
[(V,b),(V,b)] is V15() Element of [:V,V:]
{(V,b),(V,b)} is non empty set
{(V,b)} is non empty set
{{(V,b),(V,b)},{(V,b)}} is non empty set
[[(V,b),(V,b)],(b `2)] is V15() set
{[(V,b),(V,b)],(b `2)} is non empty set
{[(V,b),(V,b)]} is Relation-like Function-like non empty set
{{[(V,b),(V,b)],(b `2)},{[(V,b),(V,b)]}} is non empty set
(V,(V,b),(V,b)) is set
{ [[(V,b),(V,b)],b₁] where b₁ is Relation-like Function-like Element of (V) : [[(V,b),(V,b)],b₁] in (V) } is set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
C is Element of V
a is Element of V
(V,C,a) is set
[C,a] is V15() Element of [:V,V:]
{C,a} is non empty set
{C} is non empty set
{{C,a},{C}} is non empty set
{ [[C,a],b₁] where b₁ is Relation-like Function-like Element of (V) : [[C,a],b₁] in (V) } is set
Funcs (C,a) is functional set
b is Element of (V)
b `2 is Relation-like Function-like set
[[C,a],(b `2)] is V15() set
{[C,a],(b `2)} is non empty set
{[C,a]} is Relation-like Function-like non empty set
{{[C,a],(b `2)},{[C,a]}} is non empty set
[:C,a:] is Relation-like set
bool [:C,a:] is non empty set
V is non empty set
bool V is non empty set
C is non empty Element of bool V
a is Element of C
b is Element of C
(C,a,b) is set
(C) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of C : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of C : verum } is set
[:C,C:] is Relation-like non empty set
[a,b] is V15() Element of [:C,C:]
{a,b} is non empty set
{a} is non empty set
{{a,b},{a}} is non empty set
(C) is non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of C, b₃ is Relation-like Function-like Element of (C) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
{ [[a,b],b₁] where b₁ is Relation-like Function-like Element of (C) : [[a,b],b₁] in (C) } is set
A is Element of V
d is Element of V
(V,A,d) is set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
[A,d] is V15() Element of [:V,V:]
{A,d} is non empty set
{A} is non empty set
{{A,d},{A}} is non empty set
(V) is non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
{ [[A,d],b₁] where b₁ is Relation-like Function-like Element of (V) : [[A,d],b₁] in (V) } is set
g9 is set
f is Element of (C)
(C,f) is Element of C
f `1 is set
(f `1) `1 is set
(C,f) is Element of C
(f `1) `2 is set
[(C,f),(C,f)] is V15() Element of [:C,C:]
{(C,f),(C,f)} is non empty set
{(C,f)} is non empty set
{{(C,f),(C,f)},{(C,f)}} is non empty set
f `2 is Relation-like Function-like set
[[(C,f),(C,f)],(f `2)] is V15() set
{[(C,f),(C,f)],(f `2)} is non empty set
{[(C,f),(C,f)]} is Relation-like Function-like non empty set
{{[(C,f),(C,f)],(f `2)},{[(C,f),(C,f)]}} is non empty set
f9 is Element of (V)
(V,f9) is Element of V
f9 `1 is set
(f9 `1) `1 is set
(V,f9) is Element of V
(f9 `1) `2 is set
[[a,b],(f `2)] is V15() set
{[a,b],(f `2)} is non empty set
{[a,b]} is Relation-like Function-like non empty set
{{[a,b],(f `2)},{[a,b]}} is non empty set
f is Element of (V)
f `2 is Relation-like Function-like set
[[A,d],(f `2)] is V15() set
{[A,d],(f `2)} is non empty set
{[A,d]} is Relation-like Function-like non empty set
{{[A,d],(f `2)},{[A,d]}} is non empty set
[:A,d:] is Relation-like set
bool [:A,d:] is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
C is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
C is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
a is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
b is Element of (V)
C . b is Element of V
(V,b) is Element of V
b `1 is set
(b `1) `1 is set
a . b is Element of V
C is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
C is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
a is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
b is Element of (V)
C . b is Element of V
(V,b) is Element of V
b `1 is set
(b `1) `2 is set
a . b is Element of V
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
C is set
a is set
b is set
A is set
g9 is Element of (V)
d is Element of (V)
(V,g9,d) is Element of (V)
C is set
a is set
b is set
d is Element of (V)
A is Element of (V)
(V,d,A) is Element of (V)
C is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
proj1 C is Relation-like set
a is Element of (V)
(V,a) is Element of V
a `1 is set
(a `1) `1 is set
b is Element of (V)
[a,b] is V15() Element of [:(V),(V):]
{a,b} is non empty set
{a} is non empty set
{{a,b},{a}} is non empty set
(V,b) is Element of V
b `1 is set
(b `1) `2 is set
A is set
(V,b,a) is Element of (V)
A is Element of (V)
d is Element of (V)
(V,A) is Element of V
A `1 is set
(A `1) `1 is set
(V,d) is Element of V
d `1 is set
(d `1) `2 is set
(V,d,A) is Element of (V)
a is Element of (V)
(V,a) is Element of V
a `1 is set
(a `1) `1 is set
b is Element of (V)
(V,b) is Element of V
b `1 is set
(b `1) `2 is set
[a,b] is V15() Element of [:(V),(V):]
{a,b} is non empty set
{a} is non empty set
{{a,b},{a}} is non empty set
C . [a,b] is set
(V,b,a) is Element of (V)
C . (a,b) is set
[a,b] is V15() set
C . [a,b] is set
A is Element of (V)
d is Element of (V)
(V,A) is Element of V
A `1 is set
(A `1) `1 is set
(V,d) is Element of V
d `1 is set
(d `1) `2 is set
(V,d,A) is Element of (V)
C is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
proj1 C is Relation-like set
a is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
proj1 a is Relation-like set
b is set
A is set
[b,A] is V15() set
{b,A} is non empty set
{b} is non empty set
{{b,A},{b}} is non empty set
d is Element of (V)
(V,d) is Element of V
d `1 is set
(d `1) `1 is set
g9 is Element of (V)
(V,g9) is Element of V
g9 `1 is set
(g9 `1) `2 is set
d is Element of (V)
(V,d) is Element of V
d `1 is set
(d `1) `1 is set
g9 is Element of (V)
(V,g9) is Element of V
g9 `1 is set
(g9 `1) `2 is set
b is Element of [:(V),(V):]
A is Element of (V)
d is Element of (V)
[A,d] is V15() Element of [:(V),(V):]
{A,d} is non empty set
{A} is non empty set
{{A,d},{A}} is non empty set
(V,A) is Element of V
A `1 is set
(A `1) `1 is set
(V,d) is Element of V
d `1 is set
(d `1) `2 is set
C . [A,d] is set
(V,d,A) is Element of (V)
C . b is set
a . b is set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
V is non empty set
(V) is non empty non void V59() strict CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
d is non empty non void V59() CatStr
the carrier' of d is non empty set
f is Element of the carrier' of d
g9 is Element of the carrier' of d
[f,g9] is V15() set
{f,g9} is non empty set
{f} is non empty set
{{f,g9},{f}} is non empty set
the Comp of d is Relation-like [: the carrier' of d, the carrier' of d:] -defined the carrier' of d -valued Function-like Element of bool [:[: the carrier' of d, the carrier' of d:], the carrier' of d:]
[: the carrier' of d, the carrier' of d:] is Relation-like non empty set
[:[: the carrier' of d, the carrier' of d:], the carrier' of d:] is Relation-like non empty set
bool [:[: the carrier' of d, the carrier' of d:], the carrier' of d:] is non empty set
proj1 the Comp of d is Relation-like set
dom f is Element of the carrier of d
the carrier of d is non empty set
the Source of d is Relation-like the carrier' of d -defined the carrier of d -valued Function-like non empty total quasi_total Element of bool [: the carrier' of d, the carrier of d:]
[: the carrier' of d, the carrier of d:] is Relation-like non empty set
bool [: the carrier' of d, the carrier of d:] is non empty set
the Source of d . f is Element of the carrier of d
cod g9 is Element of the carrier of d
the Target of d is Relation-like the carrier' of d -defined the carrier of d -valued Function-like non empty total quasi_total Element of bool [: the carrier' of d, the carrier of d:]
the Target of d . g9 is Element of the carrier of d
k is Element of (V)
(V) . k is Element of V
(V,k) is Element of V
k `1 is set
(k `1) `1 is set
f9 is Element of (V)
(V) . f9 is Element of V
(V,f9) is Element of V
f9 `1 is set
(f9 `1) `2 is set
the carrier' of d is non empty set
f is Element of the carrier' of d
dom f is Element of the carrier of d
the carrier of d is non empty set
the Source of d is Relation-like the carrier' of d -defined the carrier of d -valued Function-like non empty total quasi_total Element of bool [: the carrier' of d, the carrier of d:]
[: the carrier' of d, the carrier of d:] is Relation-like non empty set
bool [: the carrier' of d, the carrier of d:] is non empty set
the Source of d . f is Element of the carrier of d
g9 is Element of the carrier' of d
cod g9 is Element of the carrier of d
the Target of d is Relation-like the carrier' of d -defined the carrier of d -valued Function-like non empty total quasi_total Element of bool [: the carrier' of d, the carrier of d:]
the Target of d . g9 is Element of the carrier of d
f (*) g9 is Element of the carrier' of d
dom (f (*) g9) is Element of the carrier of d
the Source of d . (f (*) g9) is Element of the carrier of d
dom g9 is Element of the carrier of d
the Source of d . g9 is Element of the carrier of d
cod (f (*) g9) is Element of the carrier of d
the Target of d . (f (*) g9) is Element of the carrier of d
cod f is Element of the carrier of d
the Target of d . f is Element of the carrier of d
[f,g9] is V15() Element of [: the carrier' of d, the carrier' of d:]
[: the carrier' of d, the carrier' of d:] is Relation-like non empty set
{f,g9} is non empty set
{f} is non empty set
{{f,g9},{f}} is non empty set
the Comp of d is Relation-like [: the carrier' of d, the carrier' of d:] -defined the carrier' of d -valued Function-like Element of bool [:[: the carrier' of d, the carrier' of d:], the carrier' of d:]
[:[: the carrier' of d, the carrier' of d:], the carrier' of d:] is Relation-like non empty set
bool [:[: the carrier' of d, the carrier' of d:], the carrier' of d:] is non empty set
proj1 the Comp of d is Relation-like set
the Comp of d . (f,g9) is set
[f,g9] is V15() set
the Comp of d . [f,g9] is set
k is Element of (V)
(V) . k is Element of V
(V,k) is Element of V
k `1 is set
(k `1) `1 is set
f9 is Element of (V)
(V) . f9 is Element of V
(V,f9) is Element of V
f9 `1 is set
(f9 `1) `2 is set
[k,f9] is V15() Element of [:(V),(V):]
{k,f9} is non empty set
{k} is non empty set
{{k,f9},{k}} is non empty set
(V) . [k,f9] is set
(V,f9,k) is Element of (V)
(V) . f9 is Element of V
(V,f9) is Element of V
(f9 `1) `1 is set
(V) . k is Element of V
(V,k) is Element of V
(k `1) `2 is set
(V,(V,f9,k)) is Element of V
(V,f9,k) `1 is set
((V,f9,k) `1) `1 is set
(V,(V,f9,k)) is Element of V
((V,f9,k) `1) `2 is set
the carrier' of d is non empty set
f9 is Element of the carrier' of d
dom f9 is Element of the carrier of d
the carrier of d is non empty set
the Source of d is Relation-like the carrier' of d -defined the carrier of d -valued Function-like non empty total quasi_total Element of bool [: the carrier' of d, the carrier of d:]
[: the carrier' of d, the carrier of d:] is Relation-like non empty set
bool [: the carrier' of d, the carrier of d:] is non empty set
the Source of d . f9 is Element of the carrier of d
f is Element of the carrier' of d
cod f is Element of the carrier of d
the Target of d is Relation-like the carrier' of d -defined the carrier of d -valued Function-like non empty total quasi_total Element of bool [: the carrier' of d, the carrier of d:]
the Target of d . f is Element of the carrier of d
dom f is Element of the carrier of d
the Source of d . f is Element of the carrier of d
g9 is Element of the carrier' of d
cod g9 is Element of the carrier of d
the Target of d . g9 is Element of the carrier of d
f (*) g9 is Element of the carrier' of d
f9 (*) (f (*) g9) is Element of the carrier' of d
f9 (*) f is Element of the carrier' of d
(f9 (*) f) (*) g9 is Element of the carrier' of d
g2 is Element of (V)
(V,g2) is Element of V
g2 `1 is set
(g2 `1) `1 is set
(V) . g2 is Element of V
l is Element of (V)
(V,l) is Element of V
l `1 is set
(l `1) `2 is set
(V) . l is Element of V
(V,l,g2) is Element of (V)
(V,(V,l,g2)) is Element of V
(V,l,g2) `1 is set
((V,l,g2) `1) `1 is set
(V,l) is Element of V
(l `1) `1 is set
(V) . l is Element of V
k is Element of (V)
(V,k) is Element of V
k `1 is set
(k `1) `2 is set
(V) . k is Element of V
(V,k,l) is Element of (V)
(V,(V,k,l)) is Element of V
(V,k,l) `1 is set
((V,k,l) `1) `2 is set
[f9,f] is V15() Element of [: the carrier' of d, the carrier' of d:]
[: the carrier' of d, the carrier' of d:] is Relation-like non empty set
{f9,f} is non empty set
{f9} is non empty set
{{f9,f},{f9}} is non empty set
the Comp of d is Relation-like [: the carrier' of d, the carrier' of d:] -defined the carrier' of d -valued Function-like Element of bool [:[: the carrier' of d, the carrier' of d:], the carrier' of d:]
[:[: the carrier' of d, the carrier' of d:], the carrier' of d:] is Relation-like non empty set
bool [:[: the carrier' of d, the carrier' of d:], the carrier' of d:] is non empty set
proj1 the Comp of d is Relation-like set
the Comp of d . (f9,f) is set
[f9,f] is V15() set
the Comp of d . [f9,f] is set
dom (f9 (*) f) is Element of the carrier of d
the Source of d . (f9 (*) f) is Element of the carrier of d
[(f9 (*) f),g9] is V15() Element of [: the carrier' of d, the carrier' of d:]
{(f9 (*) f),g9} is non empty set
{(f9 (*) f)} is non empty set
{{(f9 (*) f),g9},{(f9 (*) f)}} is non empty set
[f,g9] is V15() Element of [: the carrier' of d, the carrier' of d:]
{f,g9} is non empty set
{f} is non empty set
{{f,g9},{f}} is non empty set
the Comp of d . (f,g9) is set
[f,g9] is V15() set
the Comp of d . [f,g9] is set
cod (f (*) g9) is Element of the carrier of d
the Target of d . (f (*) g9) is Element of the carrier of d
[f9,(f (*) g9)] is V15() Element of [: the carrier' of d, the carrier' of d:]
{f9,(f (*) g9)} is non empty set
{{f9,(f (*) g9)},{f9}} is non empty set
the Comp of d . (f9,( the Comp of d . (f,g9))) is set
[f9,( the Comp of d . (f,g9))] is V15() set
{f9,( the Comp of d . (f,g9))} is non empty set
{{f9,( the Comp of d . (f,g9))},{f9}} is non empty set
the Comp of d . [f9,( the Comp of d . (f,g9))] is set
[g2,(V,k,l)] is V15() Element of [:(V),(V):]
{g2,(V,k,l)} is non empty set
{g2} is non empty set
{{g2,(V,k,l)},{g2}} is non empty set
(V) . [g2,(V,k,l)] is set
(V,(V,k,l),g2) is Element of (V)
(V,k,(V,l,g2)) is Element of (V)
[(V,l,g2),k] is V15() Element of [:(V),(V):]
{(V,l,g2),k} is non empty set
{(V,l,g2)} is non empty set
{{(V,l,g2),k},{(V,l,g2)}} is non empty set
(V) . [(V,l,g2),k] is set
the Comp of d . (( the Comp of d . (f9,f)),g9) is set
[( the Comp of d . (f9,f)),g9] is V15() set
{( the Comp of d . (f9,f)),g9} is non empty set
{( the Comp of d . (f9,f))} is non empty set
{{( the Comp of d . (f9,f)),g9},{( the Comp of d . (f9,f))}} is non empty set
the Comp of d . [( the Comp of d . (f9,f)),g9] is set
the carrier of d is non empty set
g9 is Element of the carrier of d
Hom (g9,g9) is Element of bool the carrier' of d
the carrier' of d is non empty set
bool the carrier' of d is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of d : ( dom b<sub>1</sub> = g9 & cod b<sub>1</sub> = g9 ) } is set
f is Element of V
(V,f) is Element of (V)
[:f,f:] is Relation-like set
bool [:f,f:] is non empty set
[f,f] is V15() Element of [:V,V:]
{f,f} is non empty set
{f} is non empty set
{{f,f},{f}} is non empty set
id f is Relation-like f -defined f -valued Function-like one-to-one total quasi_total Element of bool [:f,f:]
[[f,f],(id f)] is V15() Element of [:[:V,V:],(bool [:f,f:]):]
[:[:V,V:],(bool [:f,f:]):] is Relation-like non empty set
{[f,f],(id f)} is non empty set
{[f,f]} is Relation-like Function-like non empty set
{{[f,f],(id f)},{[f,f]}} is non empty set
f9 is Element of the carrier' of d
cod f9 is Element of the carrier of d
the Target of d is Relation-like the carrier' of d -defined the carrier of d -valued Function-like non empty total quasi_total Element of bool [: the carrier' of d, the carrier of d:]
[: the carrier' of d, the carrier of d:] is Relation-like non empty set
bool [: the carrier' of d, the carrier of d:] is non empty set
the Target of d . f9 is Element of the carrier of d
(V,(V,f)) is Element of V
(V,f) `1 is set
((V,f) `1) `2 is set
dom f9 is Element of the carrier of d
the Source of d is Relation-like the carrier' of d -defined the carrier of d -valued Function-like non empty total quasi_total Element of bool [: the carrier' of d, the carrier of d:]
the Source of d . f9 is Element of the carrier of d
(V,(V,f)) is Element of V
((V,f) `1) `1 is set
the carrier of d is non empty set
g9 is Element of the carrier of d
the carrier' of d is non empty set
f is Element of V
(V,f) is Element of (V)
[:f,f:] is Relation-like set
bool [:f,f:] is non empty set
[f,f] is V15() Element of [:V,V:]
{f,f} is non empty set
{f} is non empty set
{{f,f},{f}} is non empty set
id f is Relation-like f -defined f -valued Function-like one-to-one total quasi_total Element of bool [:f,f:]
[[f,f],(id f)] is V15() Element of [:[:V,V:],(bool [:f,f:]):]
[:[:V,V:],(bool [:f,f:]):] is Relation-like non empty set
{[f,f],(id f)} is non empty set
{[f,f]} is Relation-like Function-like non empty set
{{[f,f],(id f)},{[f,f]}} is non empty set
f9 is Element of the carrier' of d
cod f9 is Element of the carrier of d
the Target of d is Relation-like the carrier' of d -defined the carrier of d -valued Function-like non empty total quasi_total Element of bool [: the carrier' of d, the carrier of d:]
[: the carrier' of d, the carrier of d:] is Relation-like non empty set
bool [: the carrier' of d, the carrier of d:] is non empty set
the Target of d . f9 is Element of the carrier of d
(V,(V,f)) is Element of V
(V,f) `1 is set
((V,f) `1) `2 is set
dom f9 is Element of the carrier of d
the Source of d is Relation-like the carrier' of d -defined the carrier of d -valued Function-like non empty total quasi_total Element of bool [: the carrier' of d, the carrier of d:]
the Source of d . f9 is Element of the carrier of d
(V,(V,f)) is Element of V
((V,f) `1) `1 is set
k is Morphism of g9,g9
l is Element of the carrier of d
Hom (g9,l) is Element of bool the carrier' of d
bool the carrier' of d is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of d : ( dom b<sub>1</sub> = g9 & cod b<sub>1</sub> = l ) } is set
Hom (l,g9) is Element of bool the carrier' of d
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of d : ( dom b<sub>1</sub> = l & cod b<sub>1</sub> = g9 ) } is set
g2 is Morphism of g9,l
g2 (*) k is Element of the carrier' of d
g2 is Element of (V)
(V,g2) is Element of V
g2 `1 is set
(g2 `1) `1 is set
dom g2 is Element of the carrier of d
the Source of d . g2 is Element of the carrier of d
[g2,k] is V15() Element of [: the carrier' of d, the carrier' of d:]
[: the carrier' of d, the carrier' of d:] is Relation-like non empty set
{g2,k} is non empty set
{g2} is non empty set
{{g2,k},{g2}} is non empty set
proj1 (V) is Relation-like set
(V) . (g2,k) is set
[g2,k] is V15() set
(V) . [g2,k] is set
(V,(V,f),g2) is Element of (V)
g2 is Morphism of l,g9
k (*) g2 is Element of the carrier' of d
g2 is Element of (V)
(V,g2) is Element of V
g2 `1 is set
(g2 `1) `2 is set
cod g2 is Element of the carrier of d
the Target of d . g2 is Element of the carrier of d
[k,g2] is V15() Element of [: the carrier' of d, the carrier' of d:]
[: the carrier' of d, the carrier' of d:] is Relation-like non empty set
{k,g2} is non empty set
{k} is non empty set
{{k,g2},{k}} is non empty set
proj1 (V) is Relation-like set
(V) . (k,g2) is set
[k,g2] is V15() set
(V) . [k,g2] is set
(V,g2,(V,f)) is Element of (V)
V is non empty set
C is Element of V
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
V is non empty set
C is Element of V
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
C is Element of (V)
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
V is non empty set
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
C is Element of (V)
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
C is Element of the carrier' of (V)
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
C is Element of the carrier' of (V)
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
a is non empty set
(a) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(a) is non empty set
(a) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of a : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of a : verum } is set
[:a,a:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of a, b<sub>3</sub> is Relation-like Function-like Element of (a) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(a) is Relation-like (a) -defined a -valued Function-like non empty total quasi_total Element of bool [:(a),a:]
[:(a),a:] is Relation-like non empty set
bool [:(a),a:] is non empty set
(a) is Relation-like (a) -defined a -valued Function-like non empty total quasi_total Element of bool [:(a),a:]
(a) is Relation-like [:(a),(a):] -defined (a) -valued Function-like Element of bool [:[:(a),(a):],(a):]
[:(a),(a):] is Relation-like non empty set
[:[:(a),(a):],(a):] is Relation-like non empty set
bool [:[:(a),(a):],(a):] is non empty set
CatStr(# a,(a),(a),(a),(a) #) is non empty non void V59() strict CatStr
the carrier' of (a) is non empty set
C is Element of the carrier' of (V)
dom C is Element of the carrier of (V)
the carrier of (V) is non empty set
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Source of (V) . C is Element of the carrier of (V)
(V,C) is Element of (V)
(V,(V,C)) is Element of V
(V,C) `1 is set
((V,C) `1) `1 is set
b is Element of the carrier' of (a)
cod b is Element of the carrier of (a)
the carrier of (a) is non empty set
the Target of (a) is Relation-like the carrier' of (a) -defined the carrier of (a) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (a), the carrier of (a):]
[: the carrier' of (a), the carrier of (a):] is Relation-like non empty set
bool [: the carrier' of (a), the carrier of (a):] is non empty set
the Target of (a) . b is Element of the carrier of (a)
(a,b) is Element of (a)
(a,(a,b)) is Element of a
(a,b) `1 is set
((a,b) `1) `2 is set
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
(V,C) is Element of V
a is Element of the carrier of (V)
Hom (C,a) is Element of bool the carrier' of (V)
the carrier' of (V) is non empty set
bool the carrier' of (V) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of (V) : ( dom b<sub>1</sub> = C & cod b<sub>1</sub> = a ) } is set
(V,a) is Element of V
(V,(V,C),(V,a)) is set
[(V,C),(V,a)] is V15() Element of [:V,V:]
{(V,C),(V,a)} is non empty set
{(V,C)} is non empty set
{{(V,C),(V,a)},{(V,C)}} is non empty set
{ [[(V,C),(V,a)],b<sub>1</sub>] where b<sub>1</sub> is Relation-like Function-like Element of (V) : [[(V,C),(V,a)],b<sub>1</sub>] in (V) } is set
b is set
A is Element of the carrier' of (V)
cod A is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Target of (V) . A is Element of the carrier of (V)
(V,A) is Element of (V)
(V,(V,A)) is Element of V
(V,A) `1 is set
((V,A) `1) `2 is set
dom A is Element of the carrier of (V)
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Source of (V) . A is Element of the carrier of (V)
(V,(V,A)) is Element of V
((V,A) `1) `1 is set
A is Element of (V)
(V,A) is Element of V
A `1 is set
(A `1) `2 is set
(V,A) is Element of the carrier' of (V)
cod (V,A) is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Target of (V) . (V,A) is Element of the carrier of (V)
(V,A) is Element of V
(A `1) `1 is set
dom (V,A) is Element of the carrier of (V)
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Source of (V) . (V,A) is Element of the carrier of (V)
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
(V,C) is Element of V
a is Element of the carrier of (V)
Hom (C,a) is Element of bool the carrier' of (V)
the carrier' of (V) is non empty set
bool the carrier' of (V) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of (V) : ( dom b<sub>1</sub> = C & cod b<sub>1</sub> = a ) } is set
(V,a) is Element of V
Funcs ((V,C),(V,a)) is functional set
(V,(V,C),(V,a)) is set
[(V,C),(V,a)] is V15() Element of [:V,V:]
{(V,C),(V,a)} is non empty set
{(V,C)} is non empty set
{{(V,C),(V,a)},{(V,C)}} is non empty set
{ [[(V,C),(V,a)],b<sub>1</sub>] where b<sub>1</sub> is Relation-like Function-like Element of (V) : [[(V,C),(V,a)],b<sub>1</sub>] in (V) } is set
the Element of (V,(V,C),(V,a)) is Element of (V,(V,C),(V,a))
A is Element of (V)
A `2 is Relation-like Function-like set
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
C is Element of the carrier' of (V)
dom C is Element of the carrier of (V)
the carrier of (V) is non empty set
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Source of (V) . C is Element of the carrier of (V)
(V,C) is Element of (V)
a is Element of the carrier' of (V)
cod a is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Target of (V) . a is Element of the carrier of (V)
C (*) a is Element of the carrier' of (V)
(V,a) is Element of (V)
(V,(V,a),(V,C)) is Element of (V)
(V,(V,C)) is Element of V
(V,C) `1 is set
((V,C) `1) `1 is set
(V,(V,a)) is Element of V
(V,a) `1 is set
((V,a) `1) `2 is set
the Comp of (V) is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined the carrier' of (V) -valued Function-like Element of bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):]
[: the carrier' of (V), the carrier' of (V):] is Relation-like non empty set
[:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is non empty set
the Comp of (V) . ((V,C),a) is set
[(V,C),a] is V15() set
{(V,C),a} is non empty set
{(V,C)} is non empty set
{{(V,C),a},{(V,C)}} is non empty set
the Comp of (V) . [(V,C),a] is set
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
id C is Morphism of C,C
(V,C) is Element of V
(V,(V,C)) is Element of (V)
[:(V,C),(V,C):] is Relation-like set
bool [:(V,C),(V,C):] is non empty set
[(V,C),(V,C)] is V15() Element of [:V,V:]
{(V,C),(V,C)} is non empty set
{(V,C)} is non empty set
{{(V,C),(V,C)},{(V,C)}} is non empty set
id (V,C) is Relation-like (V,C) -defined (V,C) -valued Function-like one-to-one total quasi_total Element of bool [:(V,C),(V,C):]
[[(V,C),(V,C)],(id (V,C))] is V15() Element of [:[:V,V:],(bool [:(V,C),(V,C):]):]
[:[:V,V:],(bool [:(V,C),(V,C):]):] is Relation-like non empty set
{[(V,C),(V,C)],(id (V,C))} is non empty set
{[(V,C),(V,C)]} is Relation-like Function-like non empty set
{{[(V,C),(V,C)],(id (V,C))},{[(V,C),(V,C)]}} is non empty set
the carrier' of (V) is non empty set
a is Element of V
(V,a) is Element of (V)
[:a,a:] is Relation-like set
bool [:a,a:] is non empty set
[a,a] is V15() Element of [:V,V:]
{a,a} is non empty set
{a} is non empty set
{{a,a},{a}} is non empty set
id a is Relation-like a -defined a -valued Function-like one-to-one total quasi_total Element of bool [:a,a:]
[[a,a],(id a)] is V15() Element of [:[:V,V:],(bool [:a,a:]):]
[:[:V,V:],(bool [:a,a:]):] is Relation-like non empty set
{[a,a],(id a)} is non empty set
{[a,a]} is Relation-like Function-like non empty set
{{[a,a],(id a)},{[a,a]}} is non empty set
b is Element of the carrier' of (V)
cod b is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Target of (V) . b is Element of the carrier of (V)
(V,(V,a)) is Element of V
(V,a) `1 is set
((V,a) `1) `2 is set
dom b is Element of the carrier of (V)
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Source of (V) . b is Element of the carrier of (V)
(V,(V,a)) is Element of V
((V,a) `1) `1 is set
A is Morphism of C,C
d is Element of the carrier of (V)
Hom (C,d) is Element of bool the carrier' of (V)
bool the carrier' of (V) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of (V) : ( dom b<sub>1</sub> = C & cod b<sub>1</sub> = d ) } is set
Hom (d,C) is Element of bool the carrier' of (V)
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of (V) : ( dom b<sub>1</sub> = d & cod b<sub>1</sub> = C ) } is set
the Comp of (V) is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined the carrier' of (V) -valued Function-like Element of bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):]
[: the carrier' of (V), the carrier' of (V):] is Relation-like non empty set
[:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is non empty set
f is Morphism of C,d
f (*) A is Element of the carrier' of (V)
f9 is Element of (V)
(V,f9) is Element of V
f9 `1 is set
(f9 `1) `1 is set
dom f is Element of the carrier of (V)
the Source of (V) . f is Element of the carrier of (V)
[f,A] is V15() Element of the carrier' of [:(V),(V):]
[:(V),(V):] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of (V), the carrier of (V):] is Relation-like non empty set
[: the Source of (V), the Source of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is non empty set
[: the Target of (V), the Target of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
|: the Comp of (V), the Comp of (V):| is Relation-like [:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] -defined [: the carrier' of (V), the carrier' of (V):] -valued Function-like Element of bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
[:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is non empty set
CatStr(# [: the carrier of (V), the carrier of (V):],[: the carrier' of (V), the carrier' of (V):],[: the Source of (V), the Source of (V):],[: the Target of (V), the Target of (V):],|: the Comp of (V), the Comp of (V):| #) is non empty non void V59() strict CatStr
the carrier' of [:(V),(V):] is non empty set
{f,A} is non empty set
{f} is non empty set
{{f,A},{f}} is non empty set
proj1 the Comp of (V) is Relation-like set
the Comp of (V) . (f,A) is set
[f,A] is V15() set
the Comp of (V) . [f,A] is set
(V,(V,a),f9) is Element of (V)
f is Morphism of d,C
A (*) f is Element of the carrier' of (V)
f9 is Element of (V)
(V,f9) is Element of V
f9 `1 is set
(f9 `1) `2 is set
cod f is Element of the carrier of (V)
the Target of (V) . f is Element of the carrier of (V)
[A,f] is V15() Element of the carrier' of [:(V),(V):]
[:(V),(V):] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of (V), the carrier of (V):] is Relation-like non empty set
[: the Source of (V), the Source of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is non empty set
[: the Target of (V), the Target of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
|: the Comp of (V), the Comp of (V):| is Relation-like [:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] -defined [: the carrier' of (V), the carrier' of (V):] -valued Function-like Element of bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
[:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is non empty set
CatStr(# [: the carrier of (V), the carrier of (V):],[: the carrier' of (V), the carrier' of (V):],[: the Source of (V), the Source of (V):],[: the Target of (V), the Target of (V):],|: the Comp of (V), the Comp of (V):| #) is non empty non void V59() strict CatStr
the carrier' of [:(V),(V):] is non empty set
{A,f} is non empty set
{A} is non empty set
{{A,f},{A}} is non empty set
proj1 the Comp of (V) is Relation-like set
the Comp of (V) . (A,f) is set
[A,f] is V15() set
the Comp of (V) . [A,f] is set
(V,f9,(V,a)) is Element of (V)
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
a is Element of the carrier of (V)
Hom (C,a) is Element of bool the carrier' of (V)
the carrier' of (V) is non empty set
bool the carrier' of (V) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of (V) : ( dom b<sub>1</sub> = C & cod b<sub>1</sub> = a ) } is set
(V,C) is Element of V
(V,a) is Element of V
(V,(V,C),(V,a)) is set
[(V,C),(V,a)] is V15() Element of [:V,V:]
{(V,C),(V,a)} is non empty set
{(V,C)} is non empty set
{{(V,C),(V,a)},{(V,C)}} is non empty set
{ [[(V,C),(V,a)],b<sub>1</sub>] where b<sub>1</sub> is Relation-like Function-like Element of (V) : [[(V,C),(V,a)],b<sub>1</sub>] in (V) } is set
[[(V,C),(V,a)],{}] is V15() set
{[(V,C),(V,a)],{}} is non empty set
{[(V,C),(V,a)]} is Relation-like Function-like non empty set
{{[(V,C),(V,a)],{}},{[(V,C),(V,a)]}} is non empty set
[:(V,C),(V,a):] is Relation-like set
bool [:(V,C),(V,a):] is non empty set
A is Morphism of C,a
d is Morphism of C,a
g9 is Element of (V)
g9 `2 is Relation-like Function-like set
[[(V,C),(V,a)],(g9 `2)] is V15() set
{[(V,C),(V,a)],(g9 `2)} is non empty set
{{[(V,C),(V,a)],(g9 `2)},{[(V,C),(V,a)]}} is non empty set
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
a is Element of V
(V,a) is Element of the carrier of (V)
Hom (C,(V,a)) is Element of bool the carrier' of (V)
the carrier' of (V) is non empty set
bool the carrier' of (V) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of (V) : ( dom b<sub>1</sub> = C & cod b<sub>1</sub> = (V,a) ) } is set
(V,C) is Element of V
(V,(V,a)) is Element of V
Funcs ((V,C),(V,(V,a))) is functional set
V is non empty universal set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
a is set
{a} is non empty set
b is Element of the carrier of (V)
Hom (b,C) is Element of bool the carrier' of (V)
the carrier' of (V) is non empty set
bool the carrier' of (V) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of (V) : ( dom b<sub>1</sub> = b & cod b<sub>1</sub> = C ) } is set
(V,b) is Element of V
(V,C) is Element of V
[:(V,b),(V,C):] is Relation-like set
bool [:(V,b),(V,C):] is non empty set
the Relation-like (V,b) -defined (V,C) -valued Function-like quasi_total Element of bool [:(V,b),(V,C):] is Relation-like (V,b) -defined (V,C) -valued Function-like quasi_total Element of bool [:(V,b),(V,C):]
[(V,b),(V,C)] is V15() Element of [:V,V:]
{(V,b),(V,C)} is non empty set
{(V,b)} is non empty set
{{(V,b),(V,C)},{(V,b)}} is non empty set
[[(V,b),(V,C)], the Relation-like (V,b) -defined (V,C) -valued Function-like quasi_total Element of bool [:(V,b),(V,C):]] is V15() Element of [:[:V,V:],(bool [:(V,b),(V,C):]):]
[:[:V,V:],(bool [:(V,b),(V,C):]):] is Relation-like non empty set
{[(V,b),(V,C)], the Relation-like (V,b) -defined (V,C) -valued Function-like quasi_total Element of bool [:(V,b),(V,C):]} is non empty set
{[(V,b),(V,C)]} is Relation-like Function-like non empty set
{{[(V,b),(V,C)], the Relation-like (V,b) -defined (V,C) -valued Function-like quasi_total Element of bool [:(V,b),(V,C):]},{[(V,b),(V,C)]}} is non empty set
(V,(V,b),(V,C)) is set
{ [[(V,b),(V,C)],b<sub>1</sub>] where b<sub>1</sub> is Relation-like Function-like Element of (V) : [[(V,b),(V,C)],b<sub>1</sub>] in (V) } is set
g9 is Morphism of b,C
f is Morphism of b,C
f9 is Element of (V)
f9 `2 is Relation-like Function-like set
[[(V,b),(V,C)],(f9 `2)] is V15() set
{[(V,b),(V,C)],(f9 `2)} is non empty set
{{[(V,b),(V,C)],(f9 `2)},{[(V,b),(V,C)]}} is non empty set
{{}} is functional non empty set
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
(V,C) is Element of V
the Element of (V,C) is Element of (V,C)
b is set
A is Element of V
(V,A) is Element of the carrier of (V)
Hom ((V,A),C) is Element of bool the carrier' of (V)
the carrier' of (V) is non empty set
bool the carrier' of (V) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of (V) : ( dom b<sub>1</sub> = (V,A) & cod b<sub>1</sub> = C ) } is set
(V,(V,A)) is Element of V
Funcs ((V,(V,A)),(V,C)) is functional set
{ the Element of (V,C)} is non empty set
b is set
[:(V,C),(V,C):] is Relation-like set
bool [:(V,C),(V,C):] is non empty set
(V,C) --> b is Relation-like (V,C) -defined {b} -valued Function-like total quasi_total Element of bool [:(V,C),{b}:]
{b} is non empty set
[:(V,C),{b}:] is Relation-like set
bool [:(V,C),{b}:] is non empty set
(V,C) --> the Element of (V,C) is Relation-like (V,C) -defined { the Element of (V,C)} -valued Function-like total quasi_total Element of bool [:(V,C),{ the Element of (V,C)}:]
[:(V,C),{ the Element of (V,C)}:] is Relation-like set
bool [:(V,C),{ the Element of (V,C)}:] is non empty set
d is Relation-like (V,C) -defined (V,C) -valued Function-like total quasi_total Element of bool [:(V,C),(V,C):]
d . b is set
A is Relation-like (V,C) -defined (V,C) -valued Function-like total quasi_total Element of bool [:(V,C),(V,C):]
[(V,C),(V,C)] is V15() Element of [:V,V:]
{(V,C),(V,C)} is non empty set
{(V,C)} is non empty set
{{(V,C),(V,C)},{(V,C)}} is non empty set
[[(V,C),(V,C)],d] is V15() Element of [:[:V,V:],(bool [:(V,C),(V,C):]):]
[:[:V,V:],(bool [:(V,C),(V,C):]):] is Relation-like non empty set
{[(V,C),(V,C)],d} is non empty set
{[(V,C),(V,C)]} is Relation-like Function-like non empty set
{{[(V,C),(V,C)],d},{[(V,C),(V,C)]}} is non empty set
(V,(V,C),(V,C)) is set
{ [[(V,C),(V,C)],b<sub>1</sub>] where b<sub>1</sub> is Relation-like Function-like Element of (V) : [[(V,C),(V,C)],b<sub>1</sub>] in (V) } is set
[[(V,C),(V,C)],A] is V15() Element of [:[:V,V:],(bool [:(V,C),(V,C):]):]
{[(V,C),(V,C)],A} is non empty set
{{[(V,C),(V,C)],A},{[(V,C),(V,C)]}} is non empty set
f is Element of (V)
Hom (C,C) is non empty Element of bool the carrier' of (V)
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of (V) : ( dom b<sub>1</sub> = C & cod b<sub>1</sub> = C ) } is set
g9 is Element of (V)
(V,g9) is Element of the carrier' of (V)
(V,f) is Element of the carrier' of (V)
l is Morphism of C,C
f9 is Morphism of C,C
k is Morphism of C,C
V is non empty universal set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
a is Element of the carrier of (V)
Hom (C,a) is Element of bool the carrier' of (V)
the carrier' of (V) is non empty set
bool the carrier' of (V) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of (V) : ( dom b<sub>1</sub> = C & cod b<sub>1</sub> = a ) } is set
b is Morphism of C,a
(V,b) is Element of (V)
(V,b) `2 is Relation-like Function-like set
(V,(V,b)) is Element of V
(V,b) `1 is set
((V,b) `1) `1 is set
dom b is Element of the carrier of (V)
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Source of (V) . b is Element of the carrier of (V)
proj1 ((V,b) `2) is set
g9 is set
f is set
((V,b) `2) . g9 is set
((V,b) `2) . f is set
f9 is Element of V
[:f9,f9:] is Relation-like set
bool [:f9,f9:] is non empty set
f9 --> g9 is Relation-like f9 -defined {g9} -valued Function-like total quasi_total Element of bool [:f9,{g9}:]
{g9} is non empty set
[:f9,{g9}:] is Relation-like set
bool [:f9,{g9}:] is non empty set
f9 --> f is Relation-like f9 -defined {f} -valued Function-like total quasi_total Element of bool [:f9,{f}:]
{f} is non empty set
[:f9,{f}:] is Relation-like set
bool [:f9,{f}:] is non empty set
[f9,f9] is V15() Element of [:V,V:]
{f9,f9} is non empty set
{f9} is non empty set
{{f9,f9},{f9}} is non empty set
k is Relation-like f9 -defined f9 -valued Function-like total quasi_total Element of bool [:f9,f9:]
[[f9,f9],k] is V15() Element of [:[:V,V:],(bool [:f9,f9:]):]
[:[:V,V:],(bool [:f9,f9:]):] is Relation-like non empty set
{[f9,f9],k} is non empty set
{[f9,f9]} is Relation-like Function-like non empty set
{{[f9,f9],k},{[f9,f9]}} is non empty set
l is Relation-like f9 -defined f9 -valued Function-like total quasi_total Element of bool [:f9,f9:]
[[f9,f9],l] is V15() Element of [:[:V,V:],(bool [:f9,f9:]):]
{[f9,f9],l} is non empty set
{{[f9,f9],l},{[f9,f9]}} is non empty set
g2 is Element of (V)
(V,g2) is Element of the carrier' of (V)
g2 is Element of (V)
(V,g2) is Element of the carrier' of (V)
(V,(V,g2)) is Element of (V)
(V,(V,g2)) `2 is Relation-like Function-like set
((V,(V,g2)) `2) * ((V,b) `2) is Relation-like Function-like set
(V,(V,g2)) is Element of (V)
(V,(V,g2)) `2 is Relation-like Function-like set
((V,(V,g2)) `2) * ((V,b) `2) is Relation-like Function-like set
(V,(V,(V,g2)),(V,b)) is Element of (V)
(V,(V,(V,g2)),(V,b)) is Element of (V)
(V,g2) is Element of V
g2 `1 is set
(g2 `1) `1 is set
(V,g2) is Element of V
(g2 `1) `2 is set
[(V,g2),(V,g2)] is V15() Element of [:V,V:]
{(V,g2),(V,g2)} is non empty set
{(V,g2)} is non empty set
{{(V,g2),(V,g2)},{(V,g2)}} is non empty set
g2 `2 is Relation-like Function-like set
[[(V,g2),(V,g2)],(g2 `2)] is V15() set
{[(V,g2),(V,g2)],(g2 `2)} is non empty set
{[(V,g2),(V,g2)]} is Relation-like Function-like non empty set
{{[(V,g2),(V,g2)],(g2 `2)},{[(V,g2),(V,g2)]}} is non empty set
(V,(V,(V,g2)),(V,b)) `2 is Relation-like Function-like set
(V,(V,(V,(V,g2)),(V,b))) is Element of V
(V,(V,(V,g2)),(V,b)) `1 is set
((V,(V,(V,g2)),(V,b)) `1) `1 is set
(V,(V,(V,g2))) is Element of V
(V,(V,g2)) `1 is set
((V,(V,g2)) `1) `1 is set
(V,(V,(V,(V,g2)),(V,b))) is Element of V
((V,(V,(V,g2)),(V,b)) `1) `2 is set
(V,(V,b)) is Element of V
((V,b) `1) `2 is set
[(V,(V,(V,g2))),(V,(V,b))] is V15() Element of [:V,V:]
{(V,(V,(V,g2))),(V,(V,b))} is non empty set
{(V,(V,(V,g2)))} is non empty set
{{(V,(V,(V,g2))),(V,(V,b))},{(V,(V,(V,g2)))}} is non empty set
[[(V,(V,(V,g2))),(V,(V,b))],(((V,(V,g2)) `2) * ((V,b) `2))] is V15() set
{[(V,(V,(V,g2))),(V,(V,b))],(((V,(V,g2)) `2) * ((V,b) `2))} is non empty set
{[(V,(V,(V,g2))),(V,(V,b))]} is Relation-like Function-like non empty set
{{[(V,(V,(V,g2))),(V,(V,b))],(((V,(V,g2)) `2) * ((V,b) `2))},{[(V,(V,(V,g2))),(V,(V,b))]}} is non empty set
dom (V,g2) is Element of the carrier of (V)
the Source of (V) . (V,g2) is Element of the carrier of (V)
(V,(V,(V,g2))) is Element of V
((V,(V,g2)) `1) `2 is set
cod (V,g2) is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Target of (V) . (V,g2) is Element of the carrier of (V)
b (*) (V,g2) is Element of the carrier' of (V)
(V,g2) is Element of V
g2 `1 is set
(g2 `1) `1 is set
(V,g2) is Element of V
(g2 `1) `2 is set
[(V,g2),(V,g2)] is V15() Element of [:V,V:]
{(V,g2),(V,g2)} is non empty set
{(V,g2)} is non empty set
{{(V,g2),(V,g2)},{(V,g2)}} is non empty set
g2 `2 is Relation-like Function-like set
[[(V,g2),(V,g2)],(g2 `2)] is V15() set
{[(V,g2),(V,g2)],(g2 `2)} is non empty set
{[(V,g2),(V,g2)]} is Relation-like Function-like non empty set
{{[(V,g2),(V,g2)],(g2 `2)},{[(V,g2),(V,g2)]}} is non empty set
(V,(V,(V,g2)),(V,b)) `2 is Relation-like Function-like set
(V,(V,(V,(V,g2)),(V,b))) is Element of V
(V,(V,(V,g2)),(V,b)) `1 is set
((V,(V,(V,g2)),(V,b)) `1) `1 is set
(V,(V,(V,g2))) is Element of V
(V,(V,g2)) `1 is set
((V,(V,g2)) `1) `1 is set
proj1 (((V,(V,g2)) `2) * ((V,b) `2)) is set
proj1 (((V,(V,g2)) `2) * ((V,b) `2)) is set
f1f is set
((V,(V,g2)) `2) . f1f is set
(((V,(V,g2)) `2) * ((V,b) `2)) . f1f is set
((V,(V,g2)) `2) . f1f is set
(((V,(V,g2)) `2) * ((V,b) `2)) . f1f is set
(V,(V,(V,(V,g2)),(V,b))) is Element of V
((V,(V,(V,g2)),(V,b)) `1) `2 is set
[(V,(V,(V,g2))),(V,(V,b))] is V15() Element of [:V,V:]
{(V,(V,(V,g2))),(V,(V,b))} is non empty set
{(V,(V,(V,g2)))} is non empty set
{{(V,(V,(V,g2))),(V,(V,b))},{(V,(V,(V,g2)))}} is non empty set
[[(V,(V,(V,g2))),(V,(V,b))],(((V,(V,g2)) `2) * ((V,b) `2))] is V15() set
{[(V,(V,(V,g2))),(V,(V,b))],(((V,(V,g2)) `2) * ((V,b) `2))} is non empty set
{[(V,(V,(V,g2))),(V,(V,b))]} is Relation-like Function-like non empty set
{{[(V,(V,(V,g2))),(V,(V,b))],(((V,(V,g2)) `2) * ((V,b) `2))},{[(V,(V,(V,g2))),(V,(V,b))]}} is non empty set
dom (V,g2) is Element of the carrier of (V)
the Source of (V) . (V,g2) is Element of the carrier of (V)
(V,(V,(V,g2))) is Element of V
((V,(V,g2)) `1) `2 is set
cod (V,g2) is Element of the carrier of (V)
the Target of (V) . (V,g2) is Element of the carrier of (V)
f1f is Element of the carrier of (V)
Hom (f1f,C) is Element of bool the carrier' of (V)
{ b₁ where b₁ is Element of the carrier' of (V) : ( dom b₁ = f1f & cod b₁ = C ) } is set
f2f is Morphism of f1f,C
b * f2f is Morphism of f1f,a
b (*) f2f is Element of the carrier' of (V)
B is Morphism of f1f,C
b (*) B is Element of the carrier' of (V)
b * B is Morphism of f1f,a
l . g9 is set
d is Element of the carrier of (V)
Hom (d,C) is Element of bool the carrier' of (V)
{ b₁ where b₁ is Element of the carrier' of (V) : ( dom b₁ = d & cod b₁ = C ) } is set
g9 is Morphism of d,C
b * g9 is Morphism of d,a
f is Morphism of d,C
b * f is Morphism of d,a
dom g9 is Element of the carrier of (V)
the Source of (V) . g9 is Element of the carrier of (V)
dom f is Element of the carrier of (V)
the Source of (V) . f is Element of the carrier of (V)
cod g9 is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Target of (V) . g9 is Element of the carrier of (V)
cod f is Element of the carrier of (V)
the Target of (V) . f is Element of the carrier of (V)
(V,g9) is Element of (V)
(V,f) is Element of (V)
(V,(V,g9),(V,b)) is Element of (V)
b (*) g9 is Element of the carrier' of (V)
(V,(V,f),(V,b)) is Element of (V)
b (*) f is Element of the carrier' of (V)
(V,(V,g9)) is Element of V
(V,g9) `1 is set
((V,g9) `1) `1 is set
(V,(V,g9)) is Element of V
((V,g9) `1) `2 is set
[(V,(V,g9)),(V,(V,g9))] is V15() Element of [:V,V:]
{(V,(V,g9)),(V,(V,g9))} is non empty set
{(V,(V,g9))} is non empty set
{{(V,(V,g9)),(V,(V,g9))},{(V,(V,g9))}} is non empty set
(V,g9) `2 is Relation-like Function-like set
[[(V,(V,g9)),(V,(V,g9))],((V,g9) `2)] is V15() set
{[(V,(V,g9)),(V,(V,g9))],((V,g9) `2)} is non empty set
{[(V,(V,g9)),(V,(V,g9))]} is Relation-like Function-like non empty set
{{[(V,(V,g9)),(V,(V,g9))],((V,g9) `2)},{[(V,(V,g9)),(V,(V,g9))]}} is non empty set
(V,(V,f)) is Element of V
(V,f) `1 is set
((V,f) `1) `1 is set
(V,f) `2 is Relation-like Function-like set
proj1 ((V,f) `2) is set
proj2 ((V,g9) `2) is set
(V,(V,f)) is Element of V
((V,f) `1) `2 is set
proj2 ((V,f) `2) is set
(V,(V,b)) is Element of V
((V,b) `1) `2 is set
[(V,(V,f)),(V,(V,b))] is V15() Element of [:V,V:]
{(V,(V,f)),(V,(V,b))} is non empty set
{(V,(V,f))} is non empty set
{{(V,(V,f)),(V,(V,b))},{(V,(V,f))}} is non empty set
((V,f) `2) * ((V,b) `2) is Relation-like Function-like set
[[(V,(V,f)),(V,(V,b))],(((V,f) `2) * ((V,b) `2))] is V15() set
{[(V,(V,f)),(V,(V,b))],(((V,f) `2) * ((V,b) `2))} is non empty set
{[(V,(V,f)),(V,(V,b))]} is Relation-like Function-like non empty set
{{[(V,(V,f)),(V,(V,b))],(((V,f) `2) * ((V,b) `2))},{[(V,(V,f)),(V,(V,b))]}} is non empty set
[(V,(V,g9)),(V,(V,b))] is V15() Element of [:V,V:]
{(V,(V,g9)),(V,(V,b))} is non empty set
{{(V,(V,g9)),(V,(V,b))},{(V,(V,g9))}} is non empty set
((V,g9) `2) * ((V,b) `2) is Relation-like Function-like set
[[(V,(V,g9)),(V,(V,b))],(((V,g9) `2) * ((V,b) `2))] is V15() set
{[(V,(V,g9)),(V,(V,b))],(((V,g9) `2) * ((V,b) `2))} is non empty set
{[(V,(V,g9)),(V,(V,b))]} is Relation-like Function-like non empty set
{{[(V,(V,g9)),(V,(V,b))],(((V,g9) `2) * ((V,b) `2))},{[(V,(V,g9)),(V,(V,b))]}} is non empty set
proj1 ((V,g9) `2) is set
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
a is Element of the carrier of (V)
Hom (C,a) is Element of bool the carrier' of (V)
the carrier' of (V) is non empty set
bool the carrier' of (V) is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of (V) : ( dom b<sub>1</sub> = C & cod b<sub>1</sub> = a ) } is set
b is Morphism of C,a
(V,b) is Element of (V)
d is set
A is Element of V
g9 is set
(V,(V,b)) is Element of V
(V,b) `1 is set
((V,b) `1) `2 is set
(V,b) `2 is Relation-like Function-like set
proj2 ((V,b) `2) is set
[:(V,(V,b)),A:] is Relation-like set
bool [:(V,(V,b)),A:] is non empty set
(V,(V,b)) --> d is Relation-like (V,(V,b)) -defined {d} -valued Function-like total quasi_total Element of bool [:(V,(V,b)),{d}:]
{d} is non empty set
[:(V,(V,b)),{d}:] is Relation-like set
bool [:(V,(V,b)),{d}:] is non empty set
k is set
{k} is non empty set
{k} --> g9 is Relation-like {k} -defined {g9} -valued Function-like non empty total quasi_total Element of bool [:{k},{g9}:]
{g9} is non empty set
[:{k},{g9}:] is Relation-like non empty set
bool [:{k},{g9}:] is non empty set
f9 is Relation-like (V,(V,b)) -defined A -valued Function-like quasi_total Element of bool [:(V,(V,b)),A:]
f9 +* ({k} --> g9) is Relation-like Function-like set
proj1 ({k} --> g9) is non empty set
proj2 ({k} --> g9) is non empty set
proj2 f9 is set
proj2 (f9 +* ({k} --> g9)) is set
{d} \/ {g9} is non empty set
{d,g9} is non empty set
proj1 f9 is set
proj1 (f9 +* ({k} --> g9)) is set
(V,(V,b)) \/ {k} is non empty set
cod b is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Target of (V) . b is Element of the carrier of (V)
({k} --> g9) . k is set
g2 is Relation-like (V,(V,b)) -defined A -valued Function-like quasi_total Element of bool [:(V,(V,b)),A:]
g2 . k is set
f9 . k is set
[(V,(V,b)),A] is V15() Element of [:V,V:]
{(V,(V,b)),A} is non empty set
{(V,(V,b))} is non empty set
{{(V,(V,b)),A},{(V,(V,b))}} is non empty set
[[(V,(V,b)),A],f9] is V15() Element of [:[:V,V:],(bool [:(V,(V,b)),A:]):]
[:[:V,V:],(bool [:(V,(V,b)),A:]):] is Relation-like non empty set
{[(V,(V,b)),A],f9} is non empty set
{[(V,(V,b)),A]} is Relation-like Function-like non empty set
{{[(V,(V,b)),A],f9},{[(V,(V,b)),A]}} is non empty set
[[(V,(V,b)),A],g2] is V15() Element of [:[:V,V:],(bool [:(V,(V,b)),A:]):]
{[(V,(V,b)),A],g2} is non empty set
{{[(V,(V,b)),A],g2},{[(V,(V,b)),A]}} is non empty set
m1 is Element of (V)
(V,m1) is Element of the carrier' of (V)
m2 is Element of (V)
(V,m2) is Element of the carrier' of (V)
(V,(V,m1)) is Element of (V)
(V,(V,m1)) `2 is Relation-like Function-like set
((V,b) `2) * ((V,(V,m1)) `2) is Relation-like Function-like set
(V,(V,m2)) is Element of (V)
(V,(V,m2)) `2 is Relation-like Function-like set
((V,b) `2) * ((V,(V,m2)) `2) is Relation-like Function-like set
(V,(V,b),(V,(V,m1))) is Element of (V)
(V,(V,b),(V,(V,m2))) is Element of (V)
(V,m1) is Element of V
m1 `1 is set
(m1 `1) `1 is set
(V,m1) is Element of V
(m1 `1) `2 is set
[(V,m1),(V,m1)] is V15() Element of [:V,V:]
{(V,m1),(V,m1)} is non empty set
{(V,m1)} is non empty set
{{(V,m1),(V,m1)},{(V,m1)}} is non empty set
m1 `2 is Relation-like Function-like set
[[(V,m1),(V,m1)],(m1 `2)] is V15() set
{[(V,m1),(V,m1)],(m1 `2)} is non empty set
{[(V,m1),(V,m1)]} is Relation-like Function-like non empty set
{{[(V,m1),(V,m1)],(m1 `2)},{[(V,m1),(V,m1)]}} is non empty set
(V,(V,b),(V,(V,m1))) `2 is Relation-like Function-like set
(V,(V,(V,b),(V,(V,m1)))) is Element of V
(V,(V,b),(V,(V,m1))) `1 is set
((V,(V,b),(V,(V,m1))) `1) `1 is set
(V,(V,b)) is Element of V
((V,b) `1) `1 is set
(V,m2) is Element of V
m2 `1 is set
(m2 `1) `1 is set
(V,m2) is Element of V
(m2 `1) `2 is set
[(V,m2),(V,m2)] is V15() Element of [:V,V:]
{(V,m2),(V,m2)} is non empty set
{(V,m2)} is non empty set
{{(V,m2),(V,m2)},{(V,m2)}} is non empty set
m2 `2 is Relation-like Function-like set
[[(V,m2),(V,m2)],(m2 `2)] is V15() set
{[(V,m2),(V,m2)],(m2 `2)} is non empty set
{[(V,m2),(V,m2)]} is Relation-like Function-like non empty set
{{[(V,m2),(V,m2)],(m2 `2)},{[(V,m2),(V,m2)]}} is non empty set
(V,(V,b),(V,(V,m2))) `2 is Relation-like Function-like set
(V,(V,(V,b),(V,(V,m2)))) is Element of V
(V,(V,b),(V,(V,m2))) `1 is set
((V,(V,b),(V,(V,m2))) `1) `1 is set
(V,(V,(V,m2))) is Element of V
(V,(V,m2)) `1 is set
((V,(V,m2)) `1) `2 is set
cod (V,m2) is Element of the carrier of (V)
the Target of (V) . (V,m2) is Element of the carrier of (V)
(V,(V,(V,m2))) is Element of V
((V,(V,m2)) `1) `1 is set
dom (V,m2) is Element of the carrier of (V)
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Source of (V) . (V,m2) is Element of the carrier of (V)
(V,m2) (*) b is Element of the carrier' of (V)
proj1 (((V,b) `2) * ((V,(V,m1)) `2)) is set
proj1 (((V,b) `2) * ((V,(V,m2)) `2)) is set
B is set
((V,b) `2) . B is set
proj1 ((V,b) `2) is set
(((V,b) `2) * ((V,(V,m1)) `2)) . B is set
f9 . (((V,b) `2) . B) is set
(((V,b) `2) * ((V,(V,m2)) `2)) . B is set
g2 . (((V,b) `2) . B) is set
(V,(V,(V,b),(V,(V,m1)))) is Element of V
((V,(V,b),(V,(V,m1))) `1) `2 is set
(V,(V,(V,m1))) is Element of V
(V,(V,m1)) `1 is set
((V,(V,m1)) `1) `2 is set
[(V,(V,b)),(V,(V,(V,m1)))] is V15() Element of [:V,V:]
{(V,(V,b)),(V,(V,(V,m1)))} is non empty set
{(V,(V,b))} is non empty set
{{(V,(V,b)),(V,(V,(V,m1)))},{(V,(V,b))}} is non empty set
[[(V,(V,b)),(V,(V,(V,m1)))],(((V,b) `2) * ((V,(V,m1)) `2))] is V15() set
{[(V,(V,b)),(V,(V,(V,m1)))],(((V,b) `2) * ((V,(V,m1)) `2))} is non empty set
{[(V,(V,b)),(V,(V,(V,m1)))]} is Relation-like Function-like non empty set
{{[(V,(V,b)),(V,(V,(V,m1)))],(((V,b) `2) * ((V,(V,m1)) `2))},{[(V,(V,b)),(V,(V,(V,m1)))]}} is non empty set
(V,(V,(V,b),(V,(V,m2)))) is Element of V
((V,(V,b),(V,(V,m2))) `1) `2 is set
[(V,(V,b)),(V,(V,(V,m2)))] is V15() Element of [:V,V:]
{(V,(V,b)),(V,(V,(V,m2)))} is non empty set
{{(V,(V,b)),(V,(V,(V,m2)))},{(V,(V,b))}} is non empty set
[[(V,(V,b)),(V,(V,(V,m2)))],(((V,b) `2) * ((V,(V,m2)) `2))] is V15() set
{[(V,(V,b)),(V,(V,(V,m2)))],(((V,b) `2) * ((V,(V,m2)) `2))} is non empty set
{[(V,(V,b)),(V,(V,(V,m2)))]} is Relation-like Function-like non empty set
{{[(V,(V,b)),(V,(V,(V,m2)))],(((V,b) `2) * ((V,(V,m2)) `2))},{[(V,(V,b)),(V,(V,(V,m2)))]}} is non empty set
cod (V,m1) is Element of the carrier of (V)
the Target of (V) . (V,m1) is Element of the carrier of (V)
(V,(V,(V,m1))) is Element of V
((V,(V,m1)) `1) `1 is set
dom (V,m1) is Element of the carrier of (V)
the Source of (V) . (V,m1) is Element of the carrier of (V)
B is Element of the carrier of (V)
Hom (a,B) is Element of bool the carrier' of (V)
{ b₁ where b₁ is Element of the carrier' of (V) : ( dom b₁ = a & cod b₁ = B ) } is set
f1 is Morphism of a,B
f1 * b is Morphism of C,B
f1 (*) b is Element of the carrier' of (V)
f2 is Morphism of a,B
f2 (*) b is Element of the carrier' of (V)
f2 * b is Morphism of C,B
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
a is Element of the carrier of (V)
Hom (C,a) is Element of bool the carrier' of (V)
the carrier' of (V) is non empty set
bool the carrier' of (V) is non empty set
{ b₁ where b₁ is Element of the carrier' of (V) : ( dom b₁ = C & cod b₁ = a ) } is set
b is Morphism of C,a
(V,b) is Element of (V)
(V,b) `2 is Relation-like Function-like set
proj2 ((V,b) `2) is set
(V,(V,b)) is Element of V
(V,b) `1 is set
((V,b) `1) `2 is set
d is Element of the carrier of (V)
Hom (a,d) is Element of bool the carrier' of (V)
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of (V) : ( dom b<sub>1</sub> = a & cod b<sub>1</sub> = d ) } is set
g9 is Morphism of a,d
g9 * b is Morphism of C,d
f is Morphism of a,d
f * b is Morphism of C,d
dom g9 is Element of the carrier of (V)
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Source of (V) . g9 is Element of the carrier of (V)
cod b is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Target of (V) . b is Element of the carrier of (V)
dom f is Element of the carrier of (V)
the Source of (V) . f is Element of the carrier of (V)
cod g9 is Element of the carrier of (V)
the Target of (V) . g9 is Element of the carrier of (V)
cod f is Element of the carrier of (V)
the Target of (V) . f is Element of the carrier of (V)
(V,g9) is Element of (V)
(V,f) is Element of (V)
(V,(V,b),(V,g9)) is Element of (V)
g9 (*) b is Element of the carrier' of (V)
(V,(V,b),(V,f)) is Element of (V)
f (*) b is Element of the carrier' of (V)
(V,(V,g9)) is Element of V
(V,g9) `1 is set
((V,g9) `1) `1 is set
(V,(V,g9)) is Element of V
((V,g9) `1) `2 is set
[(V,(V,g9)),(V,(V,g9))] is V15() Element of [:V,V:]
{(V,(V,g9)),(V,(V,g9))} is non empty set
{(V,(V,g9))} is non empty set
{{(V,(V,g9)),(V,(V,g9))},{(V,(V,g9))}} is non empty set
(V,g9) `2 is Relation-like Function-like set
[[(V,(V,g9)),(V,(V,g9))],((V,g9) `2)] is V15() set
{[(V,(V,g9)),(V,(V,g9))],((V,g9) `2)} is non empty set
{[(V,(V,g9)),(V,(V,g9))]} is Relation-like Function-like non empty set
{{[(V,(V,g9)),(V,(V,g9))],((V,g9) `2)},{[(V,(V,g9)),(V,(V,g9))]}} is non empty set
(V,(V,f)) is Element of V
(V,f) `1 is set
((V,f) `1) `2 is set
(V,(V,f)) is Element of V
((V,f) `1) `1 is set
(V,f) `2 is Relation-like Function-like set
proj1 ((V,f) `2) is set
(V,(V,b)) is Element of V
((V,b) `1) `1 is set
[(V,(V,b)),(V,(V,f))] is V15() Element of [:V,V:]
{(V,(V,b)),(V,(V,f))} is non empty set
{(V,(V,b))} is non empty set
{{(V,(V,b)),(V,(V,f))},{(V,(V,b))}} is non empty set
((V,b) `2) * ((V,f) `2) is Relation-like Function-like set
[[(V,(V,b)),(V,(V,f))],(((V,b) `2) * ((V,f) `2))] is V15() set
{[(V,(V,b)),(V,(V,f))],(((V,b) `2) * ((V,f) `2))} is non empty set
{[(V,(V,b)),(V,(V,f))]} is Relation-like Function-like non empty set
{{[(V,(V,b)),(V,(V,f))],(((V,b) `2) * ((V,f) `2))},{[(V,(V,b)),(V,(V,f))]}} is non empty set
[(V,(V,b)),(V,(V,g9))] is V15() Element of [:V,V:]
{(V,(V,b)),(V,(V,g9))} is non empty set
{{(V,(V,b)),(V,(V,g9))},{(V,(V,b))}} is non empty set
((V,b) `2) * ((V,g9) `2) is Relation-like Function-like set
[[(V,(V,b)),(V,(V,g9))],(((V,b) `2) * ((V,g9) `2))] is V15() set
{[(V,(V,b)),(V,(V,g9))],(((V,b) `2) * ((V,g9) `2))} is non empty set
{[(V,(V,b)),(V,(V,g9))]} is Relation-like Function-like non empty set
{{[(V,(V,b)),(V,(V,g9))],(((V,b) `2) * ((V,g9) `2))},{[(V,(V,b)),(V,(V,g9))]}} is non empty set
proj1 ((V,g9) `2) is set
V is non empty universal set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is Element of the carrier of (V)
a is Element of the carrier of (V)
Hom (C,a) is Element of bool the carrier' of (V)
the carrier' of (V) is non empty set
bool the carrier' of (V) is non empty set
{ b₁ where b₁ is Element of the carrier' of (V) : ( dom b₁ = C & cod b₁ = a ) } is set
b is Morphism of C,a
(V,b) is Element of (V)
{{},{{}}} is non empty set
V is non empty set
bool V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
C is non empty Element of bool V
(C) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(C) is non empty set
(C) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of C : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of C : verum } is set
[:C,C:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of C, b₃ is Relation-like Function-like Element of (C) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(C) is Relation-like (C) -defined C -valued Function-like non empty total quasi_total Element of bool [:(C),C:]
[:(C),C:] is Relation-like non empty set
bool [:(C),C:] is non empty set
(C) is Relation-like (C) -defined C -valued Function-like non empty total quasi_total Element of bool [:(C),C:]
(C) is Relation-like [:(C),(C):] -defined (C) -valued Function-like Element of bool [:[:(C),(C):],(C):]
[:(C),(C):] is Relation-like non empty set
[:[:(C),(C):],(C):] is Relation-like non empty set
bool [:[:(C),(C):],(C):] is non empty set
CatStr(# C,(C),(C),(C),(C) #) is non empty non void V59() strict CatStr
the carrier of (C) is non empty set
the carrier of (V) is non empty set
a is Element of the carrier of (C)
b is Element of the carrier of (C)
Hom (a,b) is Element of bool the carrier' of (C)
the carrier' of (C) is non empty set
bool the carrier' of (C) is non empty set
{ b₁ where b₁ is Element of the carrier' of (C) : ( dom b₁ = a & cod b₁ = b ) } is set
A is Element of the carrier of (V)
d is Element of the carrier of (V)
Hom (A,d) is Element of bool the carrier' of (V)
the carrier' of (V) is non empty set
bool the carrier' of (V) is non empty set
{ b₁ where b₁ is Element of the carrier' of (V) : ( dom b₁ = A & cod b₁ = d ) } is set
(C,a) is Element of C
(C,b) is Element of C
(C,(C,a),(C,b)) is set
[(C,a),(C,b)] is V15() Element of [:C,C:]
{(C,a),(C,b)} is non empty set
{(C,a)} is non empty set
{{(C,a),(C,b)},{(C,a)}} is non empty set
{ [[(C,a),(C,b)],b₁] where b₁ is Relation-like Function-like Element of (C) : [[(C,a),(C,b)],b₁] in (C) } is set
(V,A) is Element of V
(V,d) is Element of V
(V,(V,A),(V,d)) is set
[(V,A),(V,d)] is V15() Element of [:V,V:]
{(V,A),(V,d)} is non empty set
{(V,A)} is non empty set
{{(V,A),(V,d)},{(V,A)}} is non empty set
{ [[(V,A),(V,d)],b₁] where b₁ is Relation-like Function-like Element of (V) : [[(V,A),(V,d)],b₁] in (V) } is set
the Comp of (C) is Relation-like [: the carrier' of (C), the carrier' of (C):] -defined the carrier' of (C) -valued Function-like Element of bool [:[: the carrier' of (C), the carrier' of (C):], the carrier' of (C):]
the carrier' of (C) is non empty set
[: the carrier' of (C), the carrier' of (C):] is Relation-like non empty set
[:[: the carrier' of (C), the carrier' of (C):], the carrier' of (C):] is Relation-like non empty set
bool [:[: the carrier' of (C), the carrier' of (C):], the carrier' of (C):] is non empty set
the Comp of (V) is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined the carrier' of (V) -valued Function-like Element of bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):]
the carrier' of (V) is non empty set
[: the carrier' of (V), the carrier' of (V):] is Relation-like non empty set
[:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is non empty set
A is Element of the carrier of (C)
g9 is Element of the carrier of (V)
d is Element of the carrier of (C)
f is Element of the carrier of (V)
Hom (A,d) is Element of bool the carrier' of (C)
bool the carrier' of (C) is non empty set
{ b₁ where b₁ is Element of the carrier' of (C) : ( dom b₁ = A & cod b₁ = d ) } is set
Hom (g9,f) is Element of bool the carrier' of (V)
bool the carrier' of (V) is non empty set
{ b₁ where b₁ is Element of the carrier' of (V) : ( dom b₁ = g9 & cod b₁ = f ) } is set
proj1 the Comp of (C) is Relation-like set
proj1 the Comp of (V) is Relation-like set
A is set
d is set
[A,d] is V15() set
{A,d} is non empty set
{A} is non empty set
{{A,d},{A}} is non empty set
g9 is Element of (C)
f is Element of (C)
[g9,f] is V15() Element of [:(C),(C):]
{g9,f} is non empty set
{g9} is non empty set
{{g9,f},{g9}} is non empty set
k is Element of (V)
(V,k) is Element of V
k `1 is set
(k `1) `1 is set
(C,g9) is Element of C
g9 `1 is set
(g9 `1) `1 is set
f9 is Element of (V)
(V,f9) is Element of V
f9 `1 is set
(f9 `1) `2 is set
(C,f) is Element of C
f `1 is set
(f `1) `2 is set
A is set
d is Element of (C)
g9 is Element of (C)
[d,g9] is V15() Element of [:(C),(C):]
{d,g9} is non empty set
{d} is non empty set
{{d,g9},{d}} is non empty set
f9 is Element of (V)
(V,f9) is Element of V
f9 `1 is set
(f9 `1) `1 is set
f is Element of (V)
(V,f) is Element of V
f `1 is set
(f `1) `2 is set
(C,d) is Element of C
d `1 is set
(d `1) `1 is set
(C,g9) is Element of C
g9 `1 is set
(g9 `1) `2 is set
(C,g9,d) is Element of (C)
(C,g9) is Element of C
(g9 `1) `1 is set
(C,d) is Element of C
(d `1) `2 is set
[(C,g9),(C,d)] is V15() Element of [:C,C:]
{(C,g9),(C,d)} is non empty set
{(C,g9)} is non empty set
{{(C,g9),(C,d)},{(C,g9)}} is non empty set
g9 `2 is Relation-like Function-like set
d `2 is Relation-like Function-like set
(g9 `2) * (d `2) is Relation-like Function-like set
[[(C,g9),(C,d)],((g9 `2) * (d `2))] is V15() set
{[(C,g9),(C,d)],((g9 `2) * (d `2))} is non empty set
{[(C,g9),(C,d)]} is Relation-like Function-like non empty set
{{[(C,g9),(C,d)],((g9 `2) * (d `2))},{[(C,g9),(C,d)]}} is non empty set
(V,f) is Element of V
(f `1) `1 is set
(V,f9) is Element of V
(f9 `1) `2 is set
(V,f,f9) is Element of (V)
the Comp of (C) . A is set
the Comp of (V) . A is set
A is Element of the carrier of (C)
id A is Morphism of A,A
d is Element of the carrier of (V)
id d is Morphism of d,d
(C,A) is Element of C
(C,(C,A)) is Element of (C)
[:(C,A),(C,A):] is Relation-like set
bool [:(C,A),(C,A):] is non empty set
[(C,A),(C,A)] is V15() Element of [:C,C:]
{(C,A),(C,A)} is non empty set
{(C,A)} is non empty set
{{(C,A),(C,A)},{(C,A)}} is non empty set
id (C,A) is Relation-like (C,A) -defined (C,A) -valued Function-like one-to-one total quasi_total Element of bool [:(C,A),(C,A):]
[[(C,A),(C,A)],(id (C,A))] is V15() Element of [:[:C,C:],(bool [:(C,A),(C,A):]):]
[:[:C,C:],(bool [:(C,A),(C,A):]):] is Relation-like non empty set
{[(C,A),(C,A)],(id (C,A))} is non empty set
{[(C,A),(C,A)]} is Relation-like Function-like non empty set
{{[(C,A),(C,A)],(id (C,A))},{[(C,A),(C,A)]}} is non empty set
(V,d) is Element of V
(V,(V,d)) is Element of (V)
[:(V,d),(V,d):] is Relation-like set
bool [:(V,d),(V,d):] is non empty set
[(V,d),(V,d)] is V15() Element of [:V,V:]
{(V,d),(V,d)} is non empty set
{(V,d)} is non empty set
{{(V,d),(V,d)},{(V,d)}} is non empty set
id (V,d) is Relation-like (V,d) -defined (V,d) -valued Function-like one-to-one total quasi_total Element of bool [:(V,d),(V,d):]
[[(V,d),(V,d)],(id (V,d))] is V15() Element of [:[:V,V:],(bool [:(V,d),(V,d):]):]
[:[:V,V:],(bool [:(V,d),(V,d):]):] is Relation-like non empty set
{[(V,d),(V,d)],(id (V,d))} is non empty set
{[(V,d),(V,d)]} is Relation-like Function-like non empty set
{{[(V,d),(V,d)],(id (V,d))},{[(V,d),(V,d)]}} is non empty set
V is non empty set
bool V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
C is non empty Element of bool V
(C) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(C) is non empty set
(C) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of C : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of C : verum } is set
[:C,C:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of C, b₃ is Relation-like Function-like Element of (C) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(C) is Relation-like (C) -defined C -valued Function-like non empty total quasi_total Element of bool [:(C),C:]
[:(C),C:] is Relation-like non empty set
bool [:(C),C:] is non empty set
(C) is Relation-like (C) -defined C -valued Function-like non empty total quasi_total Element of bool [:(C),C:]
(C) is Relation-like [:(C),(C):] -defined (C) -valued Function-like Element of bool [:[:(C),(C):],(C):]
[:(C),(C):] is Relation-like non empty set
[:[:(C),(C):],(C):] is Relation-like non empty set
bool [:[:(C),(C):],(C):] is non empty set
CatStr(# C,(C),(C),(C),(C) #) is non empty non void V59() strict CatStr
a is non empty non void V59() Category-like transitive associative reflexive with_identities Subcategory of (V)
the carrier of a is non empty set
the carrier of (V) is non empty set
b is Element of the carrier of a
A is Element of the carrier of a
Hom (b,A) is Element of bool the carrier' of a
the carrier' of a is non empty set
bool the carrier' of a is non empty set
{ b₁ where b₁ is Element of the carrier' of a : ( dom b₁ = b & cod b₁ = A ) } is set
d is Element of the carrier of (V)
g9 is Element of the carrier of (V)
Hom (d,g9) is Element of bool the carrier' of (V)
the carrier' of (V) is non empty set
bool the carrier' of (V) is non empty set
{ b₁ where b₁ is Element of the carrier' of (V) : ( dom b₁ = d & cod b₁ = g9 ) } is set
the carrier of (C) is non empty set
f is Element of the carrier of (C)
f9 is Element of the carrier of (C)
Hom (f,f9) is Element of bool the carrier' of (C)
the carrier' of (C) is non empty set
bool the carrier' of (C) is non empty set
{ b₁ where b₁ is Element of the carrier' of (C) : ( dom b₁ = f & cod b₁ = f9 ) } is set
(C,f) is Element of C
(C,f9) is Element of C
(C,(C,f),(C,f9)) is set
[(C,f),(C,f9)] is V15() Element of [:C,C:]
{(C,f),(C,f9)} is non empty set
{(C,f)} is non empty set
{{(C,f),(C,f9)},{(C,f)}} is non empty set
{ [[(C,f),(C,f9)],b₁] where b₁ is Relation-like Function-like Element of (C) : [[(C,f),(C,f9)],b₁] in (C) } is set
(V,d) is Element of V
(V,g9) is Element of V
(V,(V,d),(V,g9)) is set
[(V,d),(V,g9)] is V15() Element of [:V,V:]
{(V,d),(V,g9)} is non empty set
{(V,d)} is non empty set
{{(V,d),(V,g9)},{(V,d)}} is non empty set
{ [[(V,d),(V,g9)],b₁] where b₁ is Relation-like Function-like Element of (V) : [[(V,d),(V,g9)],b₁] in (V) } is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of V : verum } is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
(V) is set
the carrier of V is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of V : verum } is set
the Element of the carrier of V is Element of the carrier of V
Hom ( the Element of the carrier of V, the Element of the carrier of V) is non empty Element of bool the carrier' of V
the carrier' of V is non empty set
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = the Element of the carrier of V & cod b₁ = the Element of the carrier of V ) } is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
C is Element of the carrier of V
a is Element of the carrier of V
Hom (C,a) is Element of bool the carrier' of V
the carrier' of V is non empty set
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = a ) } is set
(V) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of V : verum } is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
the carrier' of V is non empty set
C is Element of the carrier of V
a is Element of the carrier' of V
cod a is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Target of V . a is Element of the carrier of V
Hom (C,(cod a)) is Element of bool the carrier' of V
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod a ) } is set
dom a is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Source of V . a is Element of the carrier of V
Hom (C,(dom a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom a ) } is set
Hom ((dom a),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = C ) } is set
Hom ((cod a),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod a & cod b₁ = C ) } is set
Hom ((dom a),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod a ) } is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
the carrier' of V is non empty set
C is Element of the carrier of V
a is Element of the carrier' of V
dom a is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Source of V . a is Element of the carrier of V
Hom (C,(dom a)) is Element of bool the carrier' of V
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom a ) } is set
cod a is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Target of V . a is Element of the carrier of V
Hom (C,(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod a ) } is set
[:(Hom (C,(dom a))),(Hom (C,(cod a))):] is Relation-like set
bool [:(Hom (C,(dom a))),(Hom (C,(cod a))):] is non empty set
d is set
g9 is Morphism of C, dom a
a (*) g9 is Element of the carrier' of V
Hom ((dom a),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod a ) } is set
f is Element of the carrier' of V
a (*) f is Element of the carrier' of V
d is Relation-like Function-like set
proj1 d is set
proj2 d is set
Hom ((dom a),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod a ) } is set
g9 is Relation-like Hom (C,(dom a)) -defined Hom (C,(cod a)) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom a))),(Hom (C,(cod a))):]
f is Element of the carrier' of V
g9 . f is set
a (*) f is Element of the carrier' of V
d is Relation-like Hom (C,(dom a)) -defined Hom (C,(cod a)) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom a))),(Hom (C,(cod a))):]
g9 is Relation-like Hom (C,(dom a)) -defined Hom (C,(cod a)) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom a))),(Hom (C,(cod a))):]
f is set
d . f is set
f9 is Element of the carrier' of V
a (*) f9 is Element of the carrier' of V
g9 . f is set
Hom ((cod a),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod a & cod b₁ = C ) } is set
Hom ((dom a),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = C ) } is set
[:(Hom ((cod a),C)),(Hom ((dom a),C)):] is Relation-like set
bool [:(Hom ((cod a),C)),(Hom ((dom a),C)):] is non empty set
d is set
g9 is Morphism of cod a,C
g9 (*) a is Element of the carrier' of V
Hom ((dom a),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod a ) } is set
f is Element of the carrier' of V
f (*) a is Element of the carrier' of V
d is Relation-like Function-like set
proj1 d is set
proj2 d is set
Hom ((dom a),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod a ) } is set
g9 is Relation-like Hom ((cod a),C) -defined Hom ((dom a),C) -valued Function-like quasi_total Element of bool [:(Hom ((cod a),C)),(Hom ((dom a),C)):]
f is Element of the carrier' of V
g9 . f is set
f (*) a is Element of the carrier' of V
d is Relation-like Hom ((cod a),C) -defined Hom ((dom a),C) -valued Function-like quasi_total Element of bool [:(Hom ((cod a),C)),(Hom ((dom a),C)):]
g9 is Relation-like Hom ((cod a),C) -defined Hom ((dom a),C) -valued Function-like quasi_total Element of bool [:(Hom ((cod a),C)),(Hom ((dom a),C)):]
f is set
d . f is set
f9 is Element of the carrier' of V
f9 (*) a is Element of the carrier' of V
g9 . f is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
C is Element of the carrier of V
a is Element of the carrier of V
id a is Morphism of a,a
(V,C,(id a)) is Relation-like Hom (C,(dom (id a))) -defined Hom (C,(cod (id a))) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom (id a)))),(Hom (C,(cod (id a)))):]
dom (id a) is Element of the carrier of V
the carrier' of V is non empty set
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Source of V . (id a) is Element of the carrier of V
Hom (C,(dom (id a))) is Element of bool the carrier' of V
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom (id a) ) } is set
cod (id a) is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Target of V . (id a) is Element of the carrier of V
Hom (C,(cod (id a))) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod (id a) ) } is set
[:(Hom (C,(dom (id a)))),(Hom (C,(cod (id a)))):] is Relation-like set
bool [:(Hom (C,(dom (id a)))),(Hom (C,(cod (id a)))):] is non empty set
Hom (C,a) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = a ) } is set
id (Hom (C,a)) is Relation-like Hom (C,a) -defined Hom (C,a) -valued Function-like one-to-one total quasi_total Element of bool [:(Hom (C,a)),(Hom (C,a)):]
[:(Hom (C,a)),(Hom (C,a)):] is Relation-like set
bool [:(Hom (C,a)),(Hom (C,a)):] is non empty set
proj1 (V,C,(id a)) is set
A is set
d is Element of the carrier' of V
cod d is Element of the carrier of V
the Target of V . d is Element of the carrier of V
(V,C,(id a)) . A is set
(id a) (*) d is Element of the carrier' of V
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
C is Element of the carrier of V
id C is Morphism of C,C
a is Element of the carrier of V
(V,a,(id C)) is Relation-like Hom ((cod (id C)),a) -defined Hom ((dom (id C)),a) -valued Function-like quasi_total Element of bool [:(Hom ((cod (id C)),a)),(Hom ((dom (id C)),a)):]
cod (id C) is Element of the carrier of V
the carrier' of V is non empty set
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Target of V . (id C) is Element of the carrier of V
Hom ((cod (id C)),a) is Element of bool the carrier' of V
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod (id C) & cod b₁ = a ) } is set
dom (id C) is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Source of V . (id C) is Element of the carrier of V
Hom ((dom (id C)),a) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom (id C) & cod b₁ = a ) } is set
[:(Hom ((cod (id C)),a)),(Hom ((dom (id C)),a)):] is Relation-like set
bool [:(Hom ((cod (id C)),a)),(Hom ((dom (id C)),a)):] is non empty set
Hom (C,a) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = a ) } is set
id (Hom (C,a)) is Relation-like Hom (C,a) -defined Hom (C,a) -valued Function-like one-to-one total quasi_total Element of bool [:(Hom (C,a)),(Hom (C,a)):]
[:(Hom (C,a)),(Hom (C,a)):] is Relation-like set
bool [:(Hom (C,a)),(Hom (C,a)):] is non empty set
proj1 (V,a,(id C)) is set
A is set
d is Element of the carrier' of V
dom d is Element of the carrier of V
the Source of V . d is Element of the carrier of V
(V,a,(id C)) . A is set
d (*) (id C) is Element of the carrier' of V
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
the carrier' of V is non empty set
C is Element of the carrier of V
a is Element of the carrier' of V
dom a is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Source of V . a is Element of the carrier of V
Hom (C,(dom a)) is Element of bool the carrier' of V
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom a ) } is set
cod a is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Target of V . a is Element of the carrier of V
Hom (C,(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod a ) } is set
(V,C,a) is Relation-like Hom (C,(dom a)) -defined Hom (C,(cod a)) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom a))),(Hom (C,(cod a))):]
[:(Hom (C,(dom a))),(Hom (C,(cod a))):] is Relation-like set
bool [:(Hom (C,(dom a))),(Hom (C,(cod a))):] is non empty set
b is Element of the carrier' of V
cod b is Element of the carrier of V
the Target of V . b is Element of the carrier of V
a (*) b is Element of the carrier' of V
(V,C,(a (*) b)) is Relation-like Hom (C,(dom (a (*) b))) -defined Hom (C,(cod (a (*) b))) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom (a (*) b)))),(Hom (C,(cod (a (*) b)))):]
dom (a (*) b) is Element of the carrier of V
the Source of V . (a (*) b) is Element of the carrier of V
Hom (C,(dom (a (*) b))) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom (a (*) b) ) } is set
cod (a (*) b) is Element of the carrier of V
the Target of V . (a (*) b) is Element of the carrier of V
Hom (C,(cod (a (*) b))) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod (a (*) b) ) } is set
[:(Hom (C,(dom (a (*) b)))),(Hom (C,(cod (a (*) b)))):] is Relation-like set
bool [:(Hom (C,(dom (a (*) b)))),(Hom (C,(cod (a (*) b)))):] is non empty set
dom b is Element of the carrier of V
the Source of V . b is Element of the carrier of V
Hom (C,(dom b)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom b ) } is set
Hom (C,(cod b)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod b ) } is set
(V,C,b) is Relation-like Hom (C,(dom b)) -defined Hom (C,(cod b)) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom b))),(Hom (C,(cod b))):]
[:(Hom (C,(dom b))),(Hom (C,(cod b))):] is Relation-like set
bool [:(Hom (C,(dom b))),(Hom (C,(cod b))):] is non empty set
(V,C,a) * (V,C,b) is Relation-like Hom (C,(dom b)) -defined Hom (C,(cod a)) -valued Function-like Element of bool [:(Hom (C,(dom b))),(Hom (C,(cod a))):]
[:(Hom (C,(dom b))),(Hom (C,(cod a))):] is Relation-like set
bool [:(Hom (C,(dom b))),(Hom (C,(cod a))):] is non empty set
Hom ((dom b),(cod b)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom b & cod b₁ = cod b ) } is set
Hom ((dom a),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod a ) } is set
proj1 (V,C,(a (*) b)) is set
proj1 ((V,C,a) * (V,C,b)) is set
f is set
f9 is Element of the carrier' of V
cod f9 is Element of the carrier of V
the Target of V . f9 is Element of the carrier of V
(V,C,b) . f is set
(V,C,(a (*) b)) . f is set
(a (*) b) (*) f9 is Element of the carrier' of V
b (*) f9 is Element of the carrier' of V
a (*) (b (*) f9) is Element of the carrier' of V
k is Element of the carrier' of V
a (*) k is Element of the carrier' of V
(V,C,a) . k is set
((V,C,a) * (V,C,b)) . f is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
the carrier' of V is non empty set
C is Element of the carrier of V
a is Element of the carrier' of V
dom a is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Source of V . a is Element of the carrier of V
cod a is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Target of V . a is Element of the carrier of V
Hom ((cod a),C) is Element of bool the carrier' of V
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod a & cod b₁ = C ) } is set
Hom ((dom a),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = C ) } is set
(V,C,a) is Relation-like Hom ((cod a),C) -defined Hom ((dom a),C) -valued Function-like quasi_total Element of bool [:(Hom ((cod a),C)),(Hom ((dom a),C)):]
[:(Hom ((cod a),C)),(Hom ((dom a),C)):] is Relation-like set
bool [:(Hom ((cod a),C)),(Hom ((dom a),C)):] is non empty set
b is Element of the carrier' of V
cod b is Element of the carrier of V
the Target of V . b is Element of the carrier of V
a (*) b is Element of the carrier' of V
(V,C,(a (*) b)) is Relation-like Hom ((cod (a (*) b)),C) -defined Hom ((dom (a (*) b)),C) -valued Function-like quasi_total Element of bool [:(Hom ((cod (a (*) b)),C)),(Hom ((dom (a (*) b)),C)):]
cod (a (*) b) is Element of the carrier of V
the Target of V . (a (*) b) is Element of the carrier of V
Hom ((cod (a (*) b)),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod (a (*) b) & cod b₁ = C ) } is set
dom (a (*) b) is Element of the carrier of V
the Source of V . (a (*) b) is Element of the carrier of V
Hom ((dom (a (*) b)),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom (a (*) b) & cod b₁ = C ) } is set
[:(Hom ((cod (a (*) b)),C)),(Hom ((dom (a (*) b)),C)):] is Relation-like set
bool [:(Hom ((cod (a (*) b)),C)),(Hom ((dom (a (*) b)),C)):] is non empty set
Hom ((cod b),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod b & cod b₁ = C ) } is set
dom b is Element of the carrier of V
the Source of V . b is Element of the carrier of V
Hom ((dom b),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom b & cod b₁ = C ) } is set
(V,C,b) is Relation-like Hom ((cod b),C) -defined Hom ((dom b),C) -valued Function-like quasi_total Element of bool [:(Hom ((cod b),C)),(Hom ((dom b),C)):]
[:(Hom ((cod b),C)),(Hom ((dom b),C)):] is Relation-like set
bool [:(Hom ((cod b),C)),(Hom ((dom b),C)):] is non empty set
(V,C,b) * (V,C,a) is Relation-like Hom ((cod a),C) -defined Hom ((dom b),C) -valued Function-like Element of bool [:(Hom ((cod a),C)),(Hom ((dom b),C)):]
[:(Hom ((cod a),C)),(Hom ((dom b),C)):] is Relation-like set
bool [:(Hom ((cod a),C)),(Hom ((dom b),C)):] is non empty set
Hom ((dom a),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod a ) } is set
Hom ((dom b),(cod b)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom b & cod b₁ = cod b ) } is set
proj1 (V,C,(a (*) b)) is set
proj1 ((V,C,b) * (V,C,a)) is set
f is set
f9 is Element of the carrier' of V
dom f9 is Element of the carrier of V
the Source of V . f9 is Element of the carrier of V
(V,C,a) . f is set
(V,C,(a (*) b)) . f is set
f9 (*) (a (*) b) is Element of the carrier' of V
f9 (*) a is Element of the carrier' of V
(f9 (*) a) (*) b is Element of the carrier' of V
k is Element of the carrier' of V
k (*) b is Element of the carrier' of V
(V,C,b) . k is set
((V,C,b) * (V,C,a)) . f is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
the carrier' of V is non empty set
bool the carrier' of V is non empty set
[:(bool the carrier' of V),(bool the carrier' of V):] is Relation-like non empty set
(V) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of V : verum } is set
((V)) is non empty set
((V)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (V) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (V) : verum } is set
[:(V),(V):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (V), b₃ is Relation-like Function-like Element of ((V)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
C is Element of the carrier of V
a is Element of the carrier' of V
dom a is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Source of V . a is Element of the carrier of V
Hom (C,(dom a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom a ) } is set
cod a is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Target of V . a is Element of the carrier of V
Hom (C,(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod a ) } is set
[:(Hom (C,(dom a))),(Hom (C,(cod a))):] is Relation-like set
bool [:(Hom (C,(dom a))),(Hom (C,(cod a))):] is non empty set
[(Hom (C,(dom a))),(Hom (C,(cod a)))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom (C,(dom a))),(Hom (C,(cod a)))} is non empty set
{(Hom (C,(dom a)))} is non empty set
{{(Hom (C,(dom a))),(Hom (C,(cod a)))},{(Hom (C,(dom a)))}} is non empty set
(V,C,a) is Relation-like Hom (C,(dom a)) -defined Hom (C,(cod a)) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom a))),(Hom (C,(cod a))):]
[[(Hom (C,(dom a))),(Hom (C,(cod a)))],(V,C,a)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom a))),(Hom (C,(cod a))):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom a))),(Hom (C,(cod a))):]):] is Relation-like non empty set
{[(Hom (C,(dom a))),(Hom (C,(cod a)))],(V,C,a)} is non empty set
{[(Hom (C,(dom a))),(Hom (C,(cod a)))]} is Relation-like Function-like non empty set
{{[(Hom (C,(dom a))),(Hom (C,(cod a)))],(V,C,a)},{[(Hom (C,(dom a))),(Hom (C,(cod a)))]}} is non empty set
Hom ((dom a),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod a ) } is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
the carrier' of V is non empty set
bool the carrier' of V is non empty set
[:(bool the carrier' of V),(bool the carrier' of V):] is Relation-like non empty set
(V) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of V : verum } is set
((V)) is non empty set
((V)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (V) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (V) : verum } is set
[:(V),(V):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (V), b₃ is Relation-like Function-like Element of ((V)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
C is Element of the carrier of V
a is Element of the carrier' of V
cod a is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Target of V . a is Element of the carrier of V
Hom ((cod a),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod a & cod b₁ = C ) } is set
dom a is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Source of V . a is Element of the carrier of V
Hom ((dom a),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = C ) } is set
[:(Hom ((cod a),C)),(Hom ((dom a),C)):] is Relation-like set
bool [:(Hom ((cod a),C)),(Hom ((dom a),C)):] is non empty set
[(Hom ((cod a),C)),(Hom ((dom a),C))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod a),C)),(Hom ((dom a),C))} is non empty set
{(Hom ((cod a),C))} is non empty set
{{(Hom ((cod a),C)),(Hom ((dom a),C))},{(Hom ((cod a),C))}} is non empty set
(V,C,a) is Relation-like Hom ((cod a),C) -defined Hom ((dom a),C) -valued Function-like quasi_total Element of bool [:(Hom ((cod a),C)),(Hom ((dom a),C)):]
[[(Hom ((cod a),C)),(Hom ((dom a),C))],(V,C,a)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod a),C)),(Hom ((dom a),C)):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod a),C)),(Hom ((dom a),C)):]):] is Relation-like non empty set
{[(Hom ((cod a),C)),(Hom ((dom a),C))],(V,C,a)} is non empty set
{[(Hom ((cod a),C)),(Hom ((dom a),C))]} is Relation-like Function-like non empty set
{{[(Hom ((cod a),C)),(Hom ((dom a),C))],(V,C,a)},{[(Hom ((cod a),C)),(Hom ((dom a),C))]}} is non empty set
Hom ((dom a),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod a ) } is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
the carrier' of V is non empty set
(V) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of V : verum } is set
((V)) is non empty set
((V)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (V) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (V) : verum } is set
[:(V),(V):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (V), b₃ is Relation-like Function-like Element of ((V)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
[: the carrier' of V,((V)):] is Relation-like non empty set
bool [: the carrier' of V,((V)):] is non empty set
bool the carrier' of V is non empty set
[:(bool the carrier' of V),(bool the carrier' of V):] is Relation-like non empty set
C is Element of the carrier of V
A is set
d is Element of the carrier' of V
dom d is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Source of V . d is Element of the carrier of V
Hom (C,(dom d)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom d ) } is set
cod d is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Target of V . d is Element of the carrier of V
Hom (C,(cod d)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod d ) } is set
[:(Hom (C,(dom d))),(Hom (C,(cod d))):] is Relation-like set
bool [:(Hom (C,(dom d))),(Hom (C,(cod d))):] is non empty set
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom d))),(Hom (C,(cod d))):]):] is Relation-like non empty set
[(Hom (C,(dom d))),(Hom (C,(cod d)))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom (C,(dom d))),(Hom (C,(cod d)))} is non empty set
{(Hom (C,(dom d)))} is non empty set
{{(Hom (C,(dom d))),(Hom (C,(cod d)))},{(Hom (C,(dom d)))}} is non empty set
(V,C,d) is Relation-like Hom (C,(dom d)) -defined Hom (C,(cod d)) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom d))),(Hom (C,(cod d))):]
[[(Hom (C,(dom d))),(Hom (C,(cod d)))],(V,C,d)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom d))),(Hom (C,(cod d))):]):]
{[(Hom (C,(dom d))),(Hom (C,(cod d)))],(V,C,d)} is non empty set
{[(Hom (C,(dom d))),(Hom (C,(cod d)))]} is Relation-like Function-like non empty set
{{[(Hom (C,(dom d))),(Hom (C,(cod d)))],(V,C,d)},{[(Hom (C,(dom d))),(Hom (C,(cod d)))]}} is non empty set
g9 is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom d))),(Hom (C,(cod d))):]):]
f is Element of the carrier' of V
dom f is Element of the carrier of V
the Source of V . f is Element of the carrier of V
Hom (C,(dom f)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom f ) } is set
cod f is Element of the carrier of V
the Target of V . f is Element of the carrier of V
Hom (C,(cod f)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod f ) } is set
[:(Hom (C,(dom f))),(Hom (C,(cod f))):] is Relation-like set
bool [:(Hom (C,(dom f))),(Hom (C,(cod f))):] is non empty set
[(Hom (C,(dom f))),(Hom (C,(cod f)))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom (C,(dom f))),(Hom (C,(cod f)))} is non empty set
{(Hom (C,(dom f)))} is non empty set
{{(Hom (C,(dom f))),(Hom (C,(cod f)))},{(Hom (C,(dom f)))}} is non empty set
(V,C,f) is Relation-like Hom (C,(dom f)) -defined Hom (C,(cod f)) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom f))),(Hom (C,(cod f))):]
[[(Hom (C,(dom f))),(Hom (C,(cod f)))],(V,C,f)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom f))),(Hom (C,(cod f))):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom f))),(Hom (C,(cod f))):]):] is Relation-like non empty set
{[(Hom (C,(dom f))),(Hom (C,(cod f)))],(V,C,f)} is non empty set
{[(Hom (C,(dom f))),(Hom (C,(cod f)))]} is Relation-like Function-like non empty set
{{[(Hom (C,(dom f))),(Hom (C,(cod f)))],(V,C,f)},{[(Hom (C,(dom f))),(Hom (C,(cod f)))]}} is non empty set
A is Relation-like Function-like set
proj1 A is set
proj2 A is set
d is Relation-like the carrier' of V -defined ((V)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V,((V)):]
g9 is Element of the carrier' of V
d . g9 is Element of ((V))
dom g9 is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Source of V . g9 is Element of the carrier of V
Hom (C,(dom g9)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom g9 ) } is set
cod g9 is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Target of V . g9 is Element of the carrier of V
Hom (C,(cod g9)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod g9 ) } is set
[:(Hom (C,(dom g9))),(Hom (C,(cod g9))):] is Relation-like set
bool [:(Hom (C,(dom g9))),(Hom (C,(cod g9))):] is non empty set
[(Hom (C,(dom g9))),(Hom (C,(cod g9)))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom (C,(dom g9))),(Hom (C,(cod g9)))} is non empty set
{(Hom (C,(dom g9)))} is non empty set
{{(Hom (C,(dom g9))),(Hom (C,(cod g9)))},{(Hom (C,(dom g9)))}} is non empty set
(V,C,g9) is Relation-like Hom (C,(dom g9)) -defined Hom (C,(cod g9)) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom g9))),(Hom (C,(cod g9))):]
[[(Hom (C,(dom g9))),(Hom (C,(cod g9)))],(V,C,g9)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom g9))),(Hom (C,(cod g9))):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom g9))),(Hom (C,(cod g9))):]):] is Relation-like non empty set
{[(Hom (C,(dom g9))),(Hom (C,(cod g9)))],(V,C,g9)} is non empty set
{[(Hom (C,(dom g9))),(Hom (C,(cod g9)))]} is Relation-like Function-like non empty set
{{[(Hom (C,(dom g9))),(Hom (C,(cod g9)))],(V,C,g9)},{[(Hom (C,(dom g9))),(Hom (C,(cod g9)))]}} is non empty set
a is Relation-like the carrier' of V -defined ((V)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V,((V)):]
b is Relation-like the carrier' of V -defined ((V)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V,((V)):]
A is Element of the carrier' of V
a . A is Element of ((V))
dom A is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Source of V . A is Element of the carrier of V
Hom (C,(dom A)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom A ) } is set
cod A is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Target of V . A is Element of the carrier of V
Hom (C,(cod A)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod A ) } is set
[:(Hom (C,(dom A))),(Hom (C,(cod A))):] is Relation-like set
bool [:(Hom (C,(dom A))),(Hom (C,(cod A))):] is non empty set
[(Hom (C,(dom A))),(Hom (C,(cod A)))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom (C,(dom A))),(Hom (C,(cod A)))} is non empty set
{(Hom (C,(dom A)))} is non empty set
{{(Hom (C,(dom A))),(Hom (C,(cod A)))},{(Hom (C,(dom A)))}} is non empty set
(V,C,A) is Relation-like Hom (C,(dom A)) -defined Hom (C,(cod A)) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom A))),(Hom (C,(cod A))):]
[[(Hom (C,(dom A))),(Hom (C,(cod A)))],(V,C,A)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom A))),(Hom (C,(cod A))):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom A))),(Hom (C,(cod A))):]):] is Relation-like non empty set
{[(Hom (C,(dom A))),(Hom (C,(cod A)))],(V,C,A)} is non empty set
{[(Hom (C,(dom A))),(Hom (C,(cod A)))]} is Relation-like Function-like non empty set
{{[(Hom (C,(dom A))),(Hom (C,(cod A)))],(V,C,A)},{[(Hom (C,(dom A))),(Hom (C,(cod A)))]}} is non empty set
b . A is Element of ((V))
A is set
d is Element of the carrier' of V
cod d is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Target of V . d is Element of the carrier of V
Hom ((cod d),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod d & cod b₁ = C ) } is set
dom d is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Source of V . d is Element of the carrier of V
Hom ((dom d),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom d & cod b₁ = C ) } is set
[:(Hom ((cod d),C)),(Hom ((dom d),C)):] is Relation-like set
bool [:(Hom ((cod d),C)),(Hom ((dom d),C)):] is non empty set
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod d),C)),(Hom ((dom d),C)):]):] is Relation-like non empty set
[(Hom ((cod d),C)),(Hom ((dom d),C))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod d),C)),(Hom ((dom d),C))} is non empty set
{(Hom ((cod d),C))} is non empty set
{{(Hom ((cod d),C)),(Hom ((dom d),C))},{(Hom ((cod d),C))}} is non empty set
(V,C,d) is Relation-like Hom ((cod d),C) -defined Hom ((dom d),C) -valued Function-like quasi_total Element of bool [:(Hom ((cod d),C)),(Hom ((dom d),C)):]
[[(Hom ((cod d),C)),(Hom ((dom d),C))],(V,C,d)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod d),C)),(Hom ((dom d),C)):]):]
{[(Hom ((cod d),C)),(Hom ((dom d),C))],(V,C,d)} is non empty set
{[(Hom ((cod d),C)),(Hom ((dom d),C))]} is Relation-like Function-like non empty set
{{[(Hom ((cod d),C)),(Hom ((dom d),C))],(V,C,d)},{[(Hom ((cod d),C)),(Hom ((dom d),C))]}} is non empty set
g9 is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod d),C)),(Hom ((dom d),C)):]):]
f is Element of the carrier' of V
cod f is Element of the carrier of V
the Target of V . f is Element of the carrier of V
Hom ((cod f),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod f & cod b₁ = C ) } is set
dom f is Element of the carrier of V
the Source of V . f is Element of the carrier of V
Hom ((dom f),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom f & cod b₁ = C ) } is set
[:(Hom ((cod f),C)),(Hom ((dom f),C)):] is Relation-like set
bool [:(Hom ((cod f),C)),(Hom ((dom f),C)):] is non empty set
[(Hom ((cod f),C)),(Hom ((dom f),C))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod f),C)),(Hom ((dom f),C))} is non empty set
{(Hom ((cod f),C))} is non empty set
{{(Hom ((cod f),C)),(Hom ((dom f),C))},{(Hom ((cod f),C))}} is non empty set
(V,C,f) is Relation-like Hom ((cod f),C) -defined Hom ((dom f),C) -valued Function-like quasi_total Element of bool [:(Hom ((cod f),C)),(Hom ((dom f),C)):]
[[(Hom ((cod f),C)),(Hom ((dom f),C))],(V,C,f)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod f),C)),(Hom ((dom f),C)):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod f),C)),(Hom ((dom f),C)):]):] is Relation-like non empty set
{[(Hom ((cod f),C)),(Hom ((dom f),C))],(V,C,f)} is non empty set
{[(Hom ((cod f),C)),(Hom ((dom f),C))]} is Relation-like Function-like non empty set
{{[(Hom ((cod f),C)),(Hom ((dom f),C))],(V,C,f)},{[(Hom ((cod f),C)),(Hom ((dom f),C))]}} is non empty set
A is Relation-like Function-like set
proj1 A is set
proj2 A is set
d is Relation-like the carrier' of V -defined ((V)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V,((V)):]
g9 is Element of the carrier' of V
d . g9 is Element of ((V))
cod g9 is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Target of V . g9 is Element of the carrier of V
Hom ((cod g9),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod g9 & cod b₁ = C ) } is set
dom g9 is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Source of V . g9 is Element of the carrier of V
Hom ((dom g9),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom g9 & cod b₁ = C ) } is set
[:(Hom ((cod g9),C)),(Hom ((dom g9),C)):] is Relation-like set
bool [:(Hom ((cod g9),C)),(Hom ((dom g9),C)):] is non empty set
[(Hom ((cod g9),C)),(Hom ((dom g9),C))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod g9),C)),(Hom ((dom g9),C))} is non empty set
{(Hom ((cod g9),C))} is non empty set
{{(Hom ((cod g9),C)),(Hom ((dom g9),C))},{(Hom ((cod g9),C))}} is non empty set
(V,C,g9) is Relation-like Hom ((cod g9),C) -defined Hom ((dom g9),C) -valued Function-like quasi_total Element of bool [:(Hom ((cod g9),C)),(Hom ((dom g9),C)):]
[[(Hom ((cod g9),C)),(Hom ((dom g9),C))],(V,C,g9)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod g9),C)),(Hom ((dom g9),C)):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod g9),C)),(Hom ((dom g9),C)):]):] is Relation-like non empty set
{[(Hom ((cod g9),C)),(Hom ((dom g9),C))],(V,C,g9)} is non empty set
{[(Hom ((cod g9),C)),(Hom ((dom g9),C))]} is Relation-like Function-like non empty set
{{[(Hom ((cod g9),C)),(Hom ((dom g9),C))],(V,C,g9)},{[(Hom ((cod g9),C)),(Hom ((dom g9),C))]}} is non empty set
a is Relation-like the carrier' of V -defined ((V)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V,((V)):]
b is Relation-like the carrier' of V -defined ((V)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V,((V)):]
A is Element of the carrier' of V
a . A is Element of ((V))
cod A is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Target of V . A is Element of the carrier of V
Hom ((cod A),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod A & cod b₁ = C ) } is set
dom A is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Source of V . A is Element of the carrier of V
Hom ((dom A),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom A & cod b₁ = C ) } is set
[:(Hom ((cod A),C)),(Hom ((dom A),C)):] is Relation-like set
bool [:(Hom ((cod A),C)),(Hom ((dom A),C)):] is non empty set
[(Hom ((cod A),C)),(Hom ((dom A),C))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod A),C)),(Hom ((dom A),C))} is non empty set
{(Hom ((cod A),C))} is non empty set
{{(Hom ((cod A),C)),(Hom ((dom A),C))},{(Hom ((cod A),C))}} is non empty set
(V,C,A) is Relation-like Hom ((cod A),C) -defined Hom ((dom A),C) -valued Function-like quasi_total Element of bool [:(Hom ((cod A),C)),(Hom ((dom A),C)):]
[[(Hom ((cod A),C)),(Hom ((dom A),C))],(V,C,A)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod A),C)),(Hom ((dom A),C)):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod A),C)),(Hom ((dom A),C)):]):] is Relation-like non empty set
{[(Hom ((cod A),C)),(Hom ((dom A),C))],(V,C,A)} is non empty set
{[(Hom ((cod A),C)),(Hom ((dom A),C))]} is Relation-like Function-like non empty set
{{[(Hom ((cod A),C)),(Hom ((dom A),C))],(V,C,A)},{[(Hom ((cod A),C)),(Hom ((dom A),C))]}} is non empty set
b . A is Element of ((V))
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier' of C is non empty set
(C) is non empty set
the carrier of C is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
[: the carrier' of C,((C)):] is Relation-like non empty set
bool [: the carrier' of C,((C)):] is non empty set
[: the carrier' of C, the carrier' of (V):] is Relation-like non empty set
bool [: the carrier' of C, the carrier' of (V):] is non empty set
a is Relation-like the carrier' of C -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C,((C)):]
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of C is non empty set
(C) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
the carrier' of C is non empty set
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
a is Element of the carrier of C
(C,a) is Relation-like the carrier' of C -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C,((C)):]
[: the carrier' of C,((C)):] is Relation-like non empty set
bool [: the carrier' of C,((C)):] is non empty set
b is Element of the carrier of C
Hom (a,b) is Element of bool the carrier' of C
bool the carrier' of C is non empty set
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = b ) } is set
id b is Morphism of b,b
(C,a) . (id b) is Element of ((C))
(C,a,(id b)) is Relation-like Hom (a,(dom (id b))) -defined Hom (a,(cod (id b))) -valued Function-like quasi_total Element of bool [:(Hom (a,(dom (id b)))),(Hom (a,(cod (id b)))):]
dom (id b) is Element of the carrier of C
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
the Source of C . (id b) is Element of the carrier of C
Hom (a,(dom (id b))) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = dom (id b) ) } is set
cod (id b) is Element of the carrier of C
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
the Target of C . (id b) is Element of the carrier of C
Hom (a,(cod (id b))) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = cod (id b) ) } is set
[:(Hom (a,(dom (id b)))),(Hom (a,(cod (id b)))):] is Relation-like set
bool [:(Hom (a,(dom (id b)))),(Hom (a,(cod (id b)))):] is non empty set
A is Element of V
id A is Relation-like A -defined A -valued Function-like one-to-one total quasi_total Element of bool [:A,A:]
[:A,A:] is Relation-like set
bool [:A,A:] is non empty set
d is Element of the carrier of (V)
id d is Morphism of d,d
(V,d) is Element of V
(V,(V,d)) is Element of (V)
[:(V,d),(V,d):] is Relation-like set
bool [:(V,d),(V,d):] is non empty set
[(V,d),(V,d)] is V15() Element of [:V,V:]
{(V,d),(V,d)} is non empty set
{(V,d)} is non empty set
{{(V,d),(V,d)},{(V,d)}} is non empty set
id (V,d) is Relation-like (V,d) -defined (V,d) -valued Function-like one-to-one total quasi_total Element of bool [:(V,d),(V,d):]
[[(V,d),(V,d)],(id (V,d))] is V15() Element of [:[:V,V:],(bool [:(V,d),(V,d):]):]
[:[:V,V:],(bool [:(V,d),(V,d):]):] is Relation-like non empty set
{[(V,d),(V,d)],(id (V,d))} is non empty set
{[(V,d),(V,d)]} is Relation-like Function-like non empty set
{{[(V,d),(V,d)],(id (V,d))},{[(V,d),(V,d)]}} is non empty set
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of C is non empty set
(C) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
the carrier' of C is non empty set
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
a is Element of the carrier of C
id a is Morphism of a,a
b is Element of the carrier of C
Hom (a,b) is Element of bool the carrier' of C
bool the carrier' of C is non empty set
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = b ) } is set
(C,b) is Relation-like the carrier' of C -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C,((C)):]
[: the carrier' of C,((C)):] is Relation-like non empty set
bool [: the carrier' of C,((C)):] is non empty set
(C,b) . (id a) is Element of ((C))
(C,b,(id a)) is Relation-like Hom ((cod (id a)),b) -defined Hom ((dom (id a)),b) -valued Function-like quasi_total Element of bool [:(Hom ((cod (id a)),b)),(Hom ((dom (id a)),b)):]
cod (id a) is Element of the carrier of C
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
the Target of C . (id a) is Element of the carrier of C
Hom ((cod (id a)),b) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = cod (id a) & cod b₁ = b ) } is set
dom (id a) is Element of the carrier of C
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
the Source of C . (id a) is Element of the carrier of C
Hom ((dom (id a)),b) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = dom (id a) & cod b₁ = b ) } is set
[:(Hom ((cod (id a)),b)),(Hom ((dom (id a)),b)):] is Relation-like set
bool [:(Hom ((cod (id a)),b)),(Hom ((dom (id a)),b)):] is non empty set
A is Element of V
id A is Relation-like A -defined A -valued Function-like one-to-one total quasi_total Element of bool [:A,A:]
[:A,A:] is Relation-like set
bool [:A,A:] is non empty set
d is Element of the carrier of (V)
id d is Morphism of d,d
(V,d) is Element of V
(V,(V,d)) is Element of (V)
[:(V,d),(V,d):] is Relation-like set
bool [:(V,d),(V,d):] is non empty set
[(V,d),(V,d)] is V15() Element of [:V,V:]
{(V,d),(V,d)} is non empty set
{(V,d)} is non empty set
{{(V,d),(V,d)},{(V,d)}} is non empty set
id (V,d) is Relation-like (V,d) -defined (V,d) -valued Function-like one-to-one total quasi_total Element of bool [:(V,d),(V,d):]
[[(V,d),(V,d)],(id (V,d))] is V15() Element of [:[:V,V:],(bool [:(V,d),(V,d):]):]
[:[:V,V:],(bool [:(V,d),(V,d):]):] is Relation-like non empty set
{[(V,d),(V,d)],(id (V,d))} is non empty set
{[(V,d),(V,d)]} is Relation-like Function-like non empty set
{{[(V,d),(V,d)],(id (V,d))},{[(V,d),(V,d)]}} is non empty set
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of C is non empty set
(C) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
the carrier' of C is non empty set
a is Element of the carrier of C
(C,a) is Relation-like the carrier' of C -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C,((C)):]
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
[: the carrier' of C,((C)):] is Relation-like non empty set
bool [: the carrier' of C,((C)):] is non empty set
[: the carrier' of C, the carrier' of (V):] is Relation-like non empty set
bool [: the carrier' of C, the carrier' of (V):] is non empty set
the carrier of (V) is non empty set
b is Relation-like the carrier' of C -defined the carrier' of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier' of (V):]
A is Element of the carrier of C
id A is Morphism of A,A
b . (id A) is Element of the carrier' of (V)
Hom (a,A) is Element of bool the carrier' of C
bool the carrier' of C is non empty set
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = A ) } is set
d is Element of V
(V,d) is Element of the carrier of (V)
g9 is Element of the carrier of (V)
id g9 is Morphism of g9,g9
A is Element of the carrier' of C
dom A is Element of the carrier of C
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
the Source of C . A is Element of the carrier of C
id (dom A) is Morphism of dom A, dom A
b . (id (dom A)) is Element of the carrier' of (V)
b . A is Element of the carrier' of (V)
dom (b . A) is Element of the carrier of (V)
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Source of (V) . (b . A) is Element of the carrier of (V)
id (dom (b . A)) is Morphism of dom (b . A), dom (b . A)
cod A is Element of the carrier of C
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
the Target of C . A is Element of the carrier of C
id (cod A) is Morphism of cod A, cod A
b . (id (cod A)) is Element of the carrier' of (V)
cod (b . A) is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Target of (V) . (b . A) is Element of the carrier of (V)
id (cod (b . A)) is Morphism of cod (b . A), cod (b . A)
Hom (a,(dom A)) is Element of bool the carrier' of C
bool the carrier' of C is non empty set
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = dom A ) } is set
Hom (a,(cod A)) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = cod A ) } is set
[:(bool the carrier' of C),(bool the carrier' of C):] is Relation-like non empty set
[:(Hom (a,(dom A))),(Hom (a,(cod A))):] is Relation-like set
bool [:(Hom (a,(dom A))),(Hom (a,(cod A))):] is non empty set
[(Hom (a,(dom A))),(Hom (a,(cod A)))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom (a,(dom A))),(Hom (a,(cod A)))} is non empty set
{(Hom (a,(dom A)))} is non empty set
{{(Hom (a,(dom A))),(Hom (a,(cod A)))},{(Hom (a,(dom A)))}} is non empty set
(C,a,A) is Relation-like Hom (a,(dom A)) -defined Hom (a,(cod A)) -valued Function-like quasi_total Element of bool [:(Hom (a,(dom A))),(Hom (a,(cod A))):]
[[(Hom (a,(dom A))),(Hom (a,(cod A)))],(C,a,A)] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom (a,(dom A))),(Hom (a,(cod A))):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom (a,(dom A))),(Hom (a,(cod A))):]):] is Relation-like non empty set
{[(Hom (a,(dom A))),(Hom (a,(cod A)))],(C,a,A)} is non empty set
{[(Hom (a,(dom A))),(Hom (a,(cod A)))]} is Relation-like Function-like non empty set
{{[(Hom (a,(dom A))),(Hom (a,(cod A)))],(C,a,A)},{[(Hom (a,(dom A))),(Hom (a,(cod A)))]}} is non empty set
(V,(b . A)) is Element of (V)
(V,(V,(b . A))) is Element of V
(V,(b . A)) `1 is set
((V,(b . A)) `1) `1 is set
(V,(V,(b . A))) is Element of V
((V,(b . A)) `1) `2 is set
[(V,(V,(b . A))),(V,(V,(b . A)))] is V15() Element of [:V,V:]
{(V,(V,(b . A))),(V,(V,(b . A)))} is non empty set
{(V,(V,(b . A)))} is non empty set
{{(V,(V,(b . A))),(V,(V,(b . A)))},{(V,(V,(b . A)))}} is non empty set
(V,(b . A)) `2 is Relation-like Function-like set
[[(V,(V,(b . A))),(V,(V,(b . A)))],((V,(b . A)) `2)] is V15() set
{[(V,(V,(b . A))),(V,(V,(b . A)))],((V,(b . A)) `2)} is non empty set
{[(V,(V,(b . A))),(V,(V,(b . A)))]} is Relation-like Function-like non empty set
{{[(V,(V,(b . A))),(V,(V,(b . A)))],((V,(b . A)) `2)},{[(V,(V,(b . A))),(V,(V,(b . A)))]}} is non empty set
[(dom (b . A)),(V,(V,(b . A)))] is V15() Element of [: the carrier of (V),V:]
[: the carrier of (V),V:] is Relation-like non empty set
{(dom (b . A)),(V,(V,(b . A)))} is non empty set
{(dom (b . A))} is non empty set
{{(dom (b . A)),(V,(V,(b . A)))},{(dom (b . A))}} is non empty set
[[(dom (b . A)),(V,(V,(b . A)))],((V,(b . A)) `2)] is V15() set
{[(dom (b . A)),(V,(V,(b . A)))],((V,(b . A)) `2)} is non empty set
{[(dom (b . A)),(V,(V,(b . A)))]} is Relation-like Function-like non empty set
{{[(dom (b . A)),(V,(V,(b . A)))],((V,(b . A)) `2)},{[(dom (b . A)),(V,(V,(b . A)))]}} is non empty set
[(dom (b . A)),(cod (b . A))] is V15() Element of the carrier of [:(V),(V):]
[:(V),(V):] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of (V), the carrier of (V):] is Relation-like non empty set
[: the carrier' of (V), the carrier' of (V):] is Relation-like non empty set
[: the Source of (V), the Source of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is non empty set
[: the Target of (V), the Target of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
the Comp of (V) is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined the carrier' of (V) -valued Function-like Element of bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):]
[:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is non empty set
|: the Comp of (V), the Comp of (V):| is Relation-like [:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] -defined [: the carrier' of (V), the carrier' of (V):] -valued Function-like Element of bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
[:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is non empty set
CatStr(# [: the carrier of (V), the carrier of (V):],[: the carrier' of (V), the carrier' of (V):],[: the Source of (V), the Source of (V):],[: the Target of (V), the Target of (V):],|: the Comp of (V), the Comp of (V):| #) is non empty non void V59() strict CatStr
the carrier of [:(V),(V):] is non empty set
{(dom (b . A)),(cod (b . A))} is non empty set
{{(dom (b . A)),(cod (b . A))},{(dom (b . A))}} is non empty set
[[(dom (b . A)),(cod (b . A))],((V,(b . A)) `2)] is V15() set
{[(dom (b . A)),(cod (b . A))],((V,(b . A)) `2)} is non empty set
{[(dom (b . A)),(cod (b . A))]} is Relation-like Function-like non empty set
{{[(dom (b . A)),(cod (b . A))],((V,(b . A)) `2)},{[(dom (b . A)),(cod (b . A))]}} is non empty set
f9 is Element of V
(V,f9) is Element of the carrier of (V)
id (V,f9) is Morphism of (V,f9),(V,f9)
k is Element of V
(V,k) is Element of the carrier of (V)
id (V,k) is Morphism of (V,k),(V,k)
d is Element of the carrier' of C
dom d is Element of the carrier of C
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
the Source of C . d is Element of the carrier of C
A is Element of the carrier' of C
cod A is Element of the carrier of C
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
the Target of C . A is Element of the carrier of C
bool the carrier' of C is non empty set
[:(bool the carrier' of C),(bool the carrier' of C):] is Relation-like non empty set
Hom (a,(dom d)) is Element of bool the carrier' of C
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of C : ( dom b<sub>1</sub> = a & cod b<sub>1</sub> = dom d ) } is set
cod d is Element of the carrier of C
the Target of C . d is Element of the carrier of C
Hom (a,(cod d)) is Element of bool the carrier' of C
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of C : ( dom b<sub>1</sub> = a & cod b<sub>1</sub> = cod d ) } is set
[:(Hom (a,(dom d))),(Hom (a,(cod d))):] is Relation-like set
bool [:(Hom (a,(dom d))),(Hom (a,(cod d))):] is non empty set
[(Hom (a,(dom d))),(Hom (a,(cod d)))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom (a,(dom d))),(Hom (a,(cod d)))} is non empty set
{(Hom (a,(dom d)))} is non empty set
{{(Hom (a,(dom d))),(Hom (a,(cod d)))},{(Hom (a,(dom d)))}} is non empty set
(C,a,d) is Relation-like Hom (a,(dom d)) -defined Hom (a,(cod d)) -valued Function-like quasi_total Element of bool [:(Hom (a,(dom d))),(Hom (a,(cod d))):]
[[(Hom (a,(dom d))),(Hom (a,(cod d)))],(C,a,d)] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom (a,(dom d))),(Hom (a,(cod d))):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom (a,(dom d))),(Hom (a,(cod d))):]):] is Relation-like non empty set
{[(Hom (a,(dom d))),(Hom (a,(cod d)))],(C,a,d)} is non empty set
{[(Hom (a,(dom d))),(Hom (a,(cod d)))]} is Relation-like Function-like non empty set
{{[(Hom (a,(dom d))),(Hom (a,(cod d)))],(C,a,d)},{[(Hom (a,(dom d))),(Hom (a,(cod d)))]}} is non empty set
b . d is Element of the carrier' of (V)
(V,(b . d)) is Element of (V)
(V,(V,(b . d))) is Element of V
(V,(b . d)) `1 is set
((V,(b . d)) `1) `1 is set
(V,(V,(b . d))) is Element of V
((V,(b . d)) `1) `2 is set
[(V,(V,(b . d))),(V,(V,(b . d)))] is V15() Element of [:V,V:]
{(V,(V,(b . d))),(V,(V,(b . d)))} is non empty set
{(V,(V,(b . d)))} is non empty set
{{(V,(V,(b . d))),(V,(V,(b . d)))},{(V,(V,(b . d)))}} is non empty set
(V,(b . d)) `2 is Relation-like Function-like set
[[(V,(V,(b . d))),(V,(V,(b . d)))],((V,(b . d)) `2)] is V15() set
{[(V,(V,(b . d))),(V,(V,(b . d)))],((V,(b . d)) `2)} is non empty set
{[(V,(V,(b . d))),(V,(V,(b . d)))]} is Relation-like Function-like non empty set
{{[(V,(V,(b . d))),(V,(V,(b . d)))],((V,(b . d)) `2)},{[(V,(V,(b . d))),(V,(V,(b . d)))]}} is non empty set
dom (b . d) is Element of the carrier of (V)
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Source of (V) . (b . d) is Element of the carrier of (V)
[(dom (b . d)),(V,(V,(b . d)))] is V15() Element of [: the carrier of (V),V:]
[: the carrier of (V),V:] is Relation-like non empty set
{(dom (b . d)),(V,(V,(b . d)))} is non empty set
{(dom (b . d))} is non empty set
{{(dom (b . d)),(V,(V,(b . d)))},{(dom (b . d))}} is non empty set
[[(dom (b . d)),(V,(V,(b . d)))],((V,(b . d)) `2)] is V15() set
{[(dom (b . d)),(V,(V,(b . d)))],((V,(b . d)) `2)} is non empty set
{[(dom (b . d)),(V,(V,(b . d)))]} is Relation-like Function-like non empty set
{{[(dom (b . d)),(V,(V,(b . d)))],((V,(b . d)) `2)},{[(dom (b . d)),(V,(V,(b . d)))]}} is non empty set
cod (b . d) is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Target of (V) . (b . d) is Element of the carrier of (V)
[(dom (b . d)),(cod (b . d))] is V15() Element of the carrier of [:(V),(V):]
[:(V),(V):] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of (V), the carrier of (V):] is Relation-like non empty set
[: the carrier' of (V), the carrier' of (V):] is Relation-like non empty set
[: the Source of (V), the Source of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is non empty set
[: the Target of (V), the Target of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
the Comp of (V) is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined the carrier' of (V) -valued Function-like Element of bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):]
[:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is non empty set
|: the Comp of (V), the Comp of (V):| is Relation-like [:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] -defined [: the carrier' of (V), the carrier' of (V):] -valued Function-like Element of bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
[:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is non empty set
CatStr(# [: the carrier of (V), the carrier of (V):],[: the carrier' of (V), the carrier' of (V):],[: the Source of (V), the Source of (V):],[: the Target of (V), the Target of (V):],|: the Comp of (V), the Comp of (V):| #) is non empty non void V59() strict CatStr
the carrier of [:(V),(V):] is non empty set
{(dom (b . d)),(cod (b . d))} is non empty set
{{(dom (b . d)),(cod (b . d))},{(dom (b . d))}} is non empty set
[[(dom (b . d)),(cod (b . d))],((V,(b . d)) `2)] is V15() set
{[(dom (b . d)),(cod (b . d))],((V,(b . d)) `2)} is non empty set
{[(dom (b . d)),(cod (b . d))]} is Relation-like Function-like non empty set
{{[(dom (b . d)),(cod (b . d))],((V,(b . d)) `2)},{[(dom (b . d)),(cod (b . d))]}} is non empty set
dom A is Element of the carrier of C
the Source of C . A is Element of the carrier of C
Hom (a,(dom A)) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = dom A ) } is set
Hom (a,(cod A)) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = cod A ) } is set
[:(Hom (a,(dom A))),(Hom (a,(cod A))):] is Relation-like set
bool [:(Hom (a,(dom A))),(Hom (a,(cod A))):] is non empty set
[(Hom (a,(dom A))),(Hom (a,(cod A)))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom (a,(dom A))),(Hom (a,(cod A)))} is non empty set
{(Hom (a,(dom A)))} is non empty set
{{(Hom (a,(dom A))),(Hom (a,(cod A)))},{(Hom (a,(dom A)))}} is non empty set
(C,a,A) is Relation-like Hom (a,(dom A)) -defined Hom (a,(cod A)) -valued Function-like quasi_total Element of bool [:(Hom (a,(dom A))),(Hom (a,(cod A))):]
[[(Hom (a,(dom A))),(Hom (a,(cod A)))],(C,a,A)] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom (a,(dom A))),(Hom (a,(cod A))):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom (a,(dom A))),(Hom (a,(cod A))):]):] is Relation-like non empty set
{[(Hom (a,(dom A))),(Hom (a,(cod A)))],(C,a,A)} is non empty set
{[(Hom (a,(dom A))),(Hom (a,(cod A)))]} is Relation-like Function-like non empty set
{{[(Hom (a,(dom A))),(Hom (a,(cod A)))],(C,a,A)},{[(Hom (a,(dom A))),(Hom (a,(cod A)))]}} is non empty set
b . A is Element of the carrier' of (V)
(V,(b . A)) is Element of (V)
(V,(V,(b . A))) is Element of V
(V,(b . A)) `1 is set
((V,(b . A)) `1) `1 is set
(V,(V,(b . A))) is Element of V
((V,(b . A)) `1) `2 is set
[(V,(V,(b . A))),(V,(V,(b . A)))] is V15() Element of [:V,V:]
{(V,(V,(b . A))),(V,(V,(b . A)))} is non empty set
{(V,(V,(b . A)))} is non empty set
{{(V,(V,(b . A))),(V,(V,(b . A)))},{(V,(V,(b . A)))}} is non empty set
(V,(b . A)) `2 is Relation-like Function-like set
[[(V,(V,(b . A))),(V,(V,(b . A)))],((V,(b . A)) `2)] is V15() set
{[(V,(V,(b . A))),(V,(V,(b . A)))],((V,(b . A)) `2)} is non empty set
{[(V,(V,(b . A))),(V,(V,(b . A)))]} is Relation-like Function-like non empty set
{{[(V,(V,(b . A))),(V,(V,(b . A)))],((V,(b . A)) `2)},{[(V,(V,(b . A))),(V,(V,(b . A)))]}} is non empty set
dom (b . A) is Element of the carrier of (V)
the Source of (V) . (b . A) is Element of the carrier of (V)
[(dom (b . A)),(V,(V,(b . A)))] is V15() Element of [: the carrier of (V),V:]
{(dom (b . A)),(V,(V,(b . A)))} is non empty set
{(dom (b . A))} is non empty set
{{(dom (b . A)),(V,(V,(b . A)))},{(dom (b . A))}} is non empty set
[[(dom (b . A)),(V,(V,(b . A)))],((V,(b . A)) `2)] is V15() set
{[(dom (b . A)),(V,(V,(b . A)))],((V,(b . A)) `2)} is non empty set
{[(dom (b . A)),(V,(V,(b . A)))]} is Relation-like Function-like non empty set
{{[(dom (b . A)),(V,(V,(b . A)))],((V,(b . A)) `2)},{[(dom (b . A)),(V,(V,(b . A)))]}} is non empty set
cod (b . A) is Element of the carrier of (V)
the Target of (V) . (b . A) is Element of the carrier of (V)
[(dom (b . A)),(cod (b . A))] is V15() Element of the carrier of [:(V),(V):]
{(dom (b . A)),(cod (b . A))} is non empty set
{{(dom (b . A)),(cod (b . A))},{(dom (b . A))}} is non empty set
[[(dom (b . A)),(cod (b . A))],((V,(b . A)) `2)] is V15() set
{[(dom (b . A)),(cod (b . A))],((V,(b . A)) `2)} is non empty set
{[(dom (b . A)),(cod (b . A))]} is Relation-like Function-like non empty set
{{[(dom (b . A)),(cod (b . A))],((V,(b . A)) `2)},{[(dom (b . A)),(cod (b . A))]}} is non empty set
d (*) A is Element of the carrier' of C
dom (d (*) A) is Element of the carrier of C
the Source of C . (d (*) A) is Element of the carrier of C
cod (d (*) A) is Element of the carrier of C
the Target of C . (d (*) A) is Element of the carrier of C
b . (d (*) A) is Element of the carrier' of (V)
Hom (a,(dom (d (*) A))) is Element of bool the carrier' of C
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of C : ( dom b<sub>1</sub> = a & cod b<sub>1</sub> = dom (d (*) A) ) } is set
Hom (a,(cod (d (*) A))) is Element of bool the carrier' of C
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of C : ( dom b<sub>1</sub> = a & cod b<sub>1</sub> = cod (d (*) A) ) } is set
[:(Hom (a,(dom (d (*) A)))),(Hom (a,(cod (d (*) A)))):] is Relation-like set
bool [:(Hom (a,(dom (d (*) A)))),(Hom (a,(cod (d (*) A)))):] is non empty set
[(Hom (a,(dom A))),(Hom (a,(cod d)))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom (a,(dom A))),(Hom (a,(cod d)))} is non empty set
{{(Hom (a,(dom A))),(Hom (a,(cod d)))},{(Hom (a,(dom A)))}} is non empty set
(C,a,(d (*) A)) is Relation-like Hom (a,(dom (d (*) A))) -defined Hom (a,(cod (d (*) A))) -valued Function-like quasi_total Element of bool [:(Hom (a,(dom (d (*) A)))),(Hom (a,(cod (d (*) A)))):]
[[(Hom (a,(dom A))),(Hom (a,(cod d)))],(C,a,(d (*) A))] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom (a,(dom (d (*) A)))),(Hom (a,(cod (d (*) A)))):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom (a,(dom (d (*) A)))),(Hom (a,(cod (d (*) A)))):]):] is Relation-like non empty set
{[(Hom (a,(dom A))),(Hom (a,(cod d)))],(C,a,(d (*) A))} is non empty set
{[(Hom (a,(dom A))),(Hom (a,(cod d)))]} is Relation-like Function-like non empty set
{{[(Hom (a,(dom A))),(Hom (a,(cod d)))],(C,a,(d (*) A))},{[(Hom (a,(dom A))),(Hom (a,(cod d)))]}} is non empty set
[:(Hom (a,(dom A))),(Hom (a,(cod d))):] is Relation-like set
bool [:(Hom (a,(dom A))),(Hom (a,(cod d))):] is non empty set
(C,a,d) * (C,a,A) is Relation-like Hom (a,(dom A)) -defined Hom (a,(cod d)) -valued Function-like Element of bool [:(Hom (a,(dom A))),(Hom (a,(cod d))):]
[[(Hom (a,(dom A))),(Hom (a,(cod d)))],((C,a,d) * (C,a,A))] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom (a,(dom A))),(Hom (a,(cod d))):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom (a,(dom A))),(Hom (a,(cod d))):]):] is Relation-like non empty set
{[(Hom (a,(dom A))),(Hom (a,(cod d)))],((C,a,d) * (C,a,A))} is non empty set
{{[(Hom (a,(dom A))),(Hom (a,(cod d)))],((C,a,d) * (C,a,A))},{[(Hom (a,(dom A))),(Hom (a,(cod d)))]}} is non empty set
(V,(V,(b . A)),(V,(b . d))) is Element of (V)
(b . d) (*) (b . A) is Element of the carrier' of (V)
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of V, b<sub>3</sub> is Relation-like Function-like Element of (V) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of C is non empty set
(C) is non empty set
{ (Hom (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of the carrier of C : verum } is set
the carrier' of C is non empty set
a is Element of the carrier of C
(C,a) is Relation-like the carrier' of C -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C,((C)):]
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of (C) : verum } is set
union { (Funcs (b<sub>1</sub>,b<sub>2</sub>)) where b<sub>1</sub>, b<sub>2</sub> is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b<sub>1</sub>,b<sub>2</sub>],b<sub>3</sub>] where b<sub>1</sub>, b<sub>2</sub> is Element of (C), b<sub>3</sub> is Relation-like Function-like Element of ((C)) : ( ( not b<sub>2</sub> = {} or b<sub>1</sub> = {} ) & b<sub>3</sub> is Relation-like b<sub>1</sub> -defined b<sub>2</sub> -valued Function-like quasi_total Element of bool [:b<sub>1</sub>,b<sub>2</sub>:] ) } is set
[: the carrier' of C,((C)):] is Relation-like non empty set
bool [: the carrier' of C,((C)):] is non empty set
[: the carrier' of C, the carrier' of (V):] is Relation-like non empty set
bool [: the carrier' of C, the carrier' of (V):] is non empty set
the carrier of (V) is non empty set
b is Relation-like the carrier' of C -defined the carrier' of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier' of (V):]
A is Element of the carrier of C
id A is Morphism of A,A
b . (id A) is Element of the carrier' of (V)
Hom (A,a) is Element of bool the carrier' of C
bool the carrier' of C is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of C : ( dom b<sub>1</sub> = A & cod b<sub>1</sub> = a ) } is set
d is Element of V
(V,d) is Element of the carrier of (V)
g9 is Element of the carrier of (V)
id g9 is Morphism of g9,g9
A is Element of the carrier' of C
dom A is Element of the carrier of C
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
the Source of C . A is Element of the carrier of C
id (dom A) is Morphism of dom A, dom A
b . (id (dom A)) is Element of the carrier' of (V)
b . A is Element of the carrier' of (V)
cod (b . A) is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Target of (V) . (b . A) is Element of the carrier of (V)
id (cod (b . A)) is Morphism of cod (b . A), cod (b . A)
cod A is Element of the carrier of C
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
the Target of C . A is Element of the carrier of C
id (cod A) is Morphism of cod A, cod A
b . (id (cod A)) is Element of the carrier' of (V)
dom (b . A) is Element of the carrier of (V)
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Source of (V) . (b . A) is Element of the carrier of (V)
id (dom (b . A)) is Morphism of dom (b . A), dom (b . A)
Hom ((cod A),a) is Element of bool the carrier' of C
bool the carrier' of C is non empty set
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of C : ( dom b<sub>1</sub> = cod A & cod b<sub>1</sub> = a ) } is set
Hom ((dom A),a) is Element of bool the carrier' of C
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of C : ( dom b<sub>1</sub> = dom A & cod b<sub>1</sub> = a ) } is set
[:(bool the carrier' of C),(bool the carrier' of C):] is Relation-like non empty set
[:(Hom ((cod A),a)),(Hom ((dom A),a)):] is Relation-like set
bool [:(Hom ((cod A),a)),(Hom ((dom A),a)):] is non empty set
[(Hom ((cod A),a)),(Hom ((dom A),a))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom ((cod A),a)),(Hom ((dom A),a))} is non empty set
{(Hom ((cod A),a))} is non empty set
{{(Hom ((cod A),a)),(Hom ((dom A),a))},{(Hom ((cod A),a))}} is non empty set
(C,a,A) is Relation-like Hom ((cod A),a) -defined Hom ((dom A),a) -valued Function-like quasi_total Element of bool [:(Hom ((cod A),a)),(Hom ((dom A),a)):]
[[(Hom ((cod A),a)),(Hom ((dom A),a))],(C,a,A)] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod A),a)),(Hom ((dom A),a)):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod A),a)),(Hom ((dom A),a)):]):] is Relation-like non empty set
{[(Hom ((cod A),a)),(Hom ((dom A),a))],(C,a,A)} is non empty set
{[(Hom ((cod A),a)),(Hom ((dom A),a))]} is Relation-like Function-like non empty set
{{[(Hom ((cod A),a)),(Hom ((dom A),a))],(C,a,A)},{[(Hom ((cod A),a)),(Hom ((dom A),a))]}} is non empty set
(V,(b . A)) is Element of (V)
(V,(V,(b . A))) is Element of V
(V,(b . A)) `1 is set
((V,(b . A)) `1) `1 is set
(V,(V,(b . A))) is Element of V
((V,(b . A)) `1) `2 is set
[(V,(V,(b . A))),(V,(V,(b . A)))] is V15() Element of [:V,V:]
{(V,(V,(b . A))),(V,(V,(b . A)))} is non empty set
{(V,(V,(b . A)))} is non empty set
{{(V,(V,(b . A))),(V,(V,(b . A)))},{(V,(V,(b . A)))}} is non empty set
(V,(b . A)) `2 is Relation-like Function-like set
[[(V,(V,(b . A))),(V,(V,(b . A)))],((V,(b . A)) `2)] is V15() set
{[(V,(V,(b . A))),(V,(V,(b . A)))],((V,(b . A)) `2)} is non empty set
{[(V,(V,(b . A))),(V,(V,(b . A)))]} is Relation-like Function-like non empty set
{{[(V,(V,(b . A))),(V,(V,(b . A)))],((V,(b . A)) `2)},{[(V,(V,(b . A))),(V,(V,(b . A)))]}} is non empty set
[(dom (b . A)),(V,(V,(b . A)))] is V15() Element of [: the carrier of (V),V:]
[: the carrier of (V),V:] is Relation-like non empty set
{(dom (b . A)),(V,(V,(b . A)))} is non empty set
{(dom (b . A))} is non empty set
{{(dom (b . A)),(V,(V,(b . A)))},{(dom (b . A))}} is non empty set
[[(dom (b . A)),(V,(V,(b . A)))],((V,(b . A)) `2)] is V15() set
{[(dom (b . A)),(V,(V,(b . A)))],((V,(b . A)) `2)} is non empty set
{[(dom (b . A)),(V,(V,(b . A)))]} is Relation-like Function-like non empty set
{{[(dom (b . A)),(V,(V,(b . A)))],((V,(b . A)) `2)},{[(dom (b . A)),(V,(V,(b . A)))]}} is non empty set
[(dom (b . A)),(cod (b . A))] is V15() Element of the carrier of [:(V),(V):]
[:(V),(V):] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of (V), the carrier of (V):] is Relation-like non empty set
[: the carrier' of (V), the carrier' of (V):] is Relation-like non empty set
[: the Source of (V), the Source of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is non empty set
[: the Target of (V), the Target of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
the Comp of (V) is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined the carrier' of (V) -valued Function-like Element of bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):]
[:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is non empty set
|: the Comp of (V), the Comp of (V):| is Relation-like [:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] -defined [: the carrier' of (V), the carrier' of (V):] -valued Function-like Element of bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
[:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is non empty set
CatStr(# [: the carrier of (V), the carrier of (V):],[: the carrier' of (V), the carrier' of (V):],[: the Source of (V), the Source of (V):],[: the Target of (V), the Target of (V):],|: the Comp of (V), the Comp of (V):| #) is non empty non void V59() strict CatStr
the carrier of [:(V),(V):] is non empty set
{(dom (b . A)),(cod (b . A))} is non empty set
{{(dom (b . A)),(cod (b . A))},{(dom (b . A))}} is non empty set
[[(dom (b . A)),(cod (b . A))],((V,(b . A)) `2)] is V15() set
{[(dom (b . A)),(cod (b . A))],((V,(b . A)) `2)} is non empty set
{[(dom (b . A)),(cod (b . A))]} is Relation-like Function-like non empty set
{{[(dom (b . A)),(cod (b . A))],((V,(b . A)) `2)},{[(dom (b . A)),(cod (b . A))]}} is non empty set
k is Element of V
(V,k) is Element of the carrier of (V)
id (V,k) is Morphism of (V,k),(V,k)
f9 is Element of V
(V,f9) is Element of the carrier of (V)
id (V,f9) is Morphism of (V,f9),(V,f9)
d is Element of the carrier' of C
dom d is Element of the carrier of C
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
the Source of C . d is Element of the carrier of C
A is Element of the carrier' of C
cod A is Element of the carrier of C
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
the Target of C . A is Element of the carrier of C
bool the carrier' of C is non empty set
[:(bool the carrier' of C),(bool the carrier' of C):] is Relation-like non empty set
cod d is Element of the carrier of C
the Target of C . d is Element of the carrier of C
Hom ((cod d),a) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = cod d & cod b₁ = a ) } is set
Hom ((dom d),a) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = dom d & cod b₁ = a ) } is set
[:(Hom ((cod d),a)),(Hom ((dom d),a)):] is Relation-like set
bool [:(Hom ((cod d),a)),(Hom ((dom d),a)):] is non empty set
[(Hom ((cod d),a)),(Hom ((dom d),a))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom ((cod d),a)),(Hom ((dom d),a))} is non empty set
{(Hom ((cod d),a))} is non empty set
{{(Hom ((cod d),a)),(Hom ((dom d),a))},{(Hom ((cod d),a))}} is non empty set
(C,a,d) is Relation-like Hom ((cod d),a) -defined Hom ((dom d),a) -valued Function-like quasi_total Element of bool [:(Hom ((cod d),a)),(Hom ((dom d),a)):]
[[(Hom ((cod d),a)),(Hom ((dom d),a))],(C,a,d)] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod d),a)),(Hom ((dom d),a)):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod d),a)),(Hom ((dom d),a)):]):] is Relation-like non empty set
{[(Hom ((cod d),a)),(Hom ((dom d),a))],(C,a,d)} is non empty set
{[(Hom ((cod d),a)),(Hom ((dom d),a))]} is Relation-like Function-like non empty set
{{[(Hom ((cod d),a)),(Hom ((dom d),a))],(C,a,d)},{[(Hom ((cod d),a)),(Hom ((dom d),a))]}} is non empty set
b . d is Element of the carrier' of (V)
(V,(b . d)) is Element of (V)
(V,(V,(b . d))) is Element of V
(V,(b . d)) `1 is set
((V,(b . d)) `1) `1 is set
(V,(V,(b . d))) is Element of V
((V,(b . d)) `1) `2 is set
[(V,(V,(b . d))),(V,(V,(b . d)))] is V15() Element of [:V,V:]
{(V,(V,(b . d))),(V,(V,(b . d)))} is non empty set
{(V,(V,(b . d)))} is non empty set
{{(V,(V,(b . d))),(V,(V,(b . d)))},{(V,(V,(b . d)))}} is non empty set
(V,(b . d)) `2 is Relation-like Function-like set
[[(V,(V,(b . d))),(V,(V,(b . d)))],((V,(b . d)) `2)] is V15() set
{[(V,(V,(b . d))),(V,(V,(b . d)))],((V,(b . d)) `2)} is non empty set
{[(V,(V,(b . d))),(V,(V,(b . d)))]} is Relation-like Function-like non empty set
{{[(V,(V,(b . d))),(V,(V,(b . d)))],((V,(b . d)) `2)},{[(V,(V,(b . d))),(V,(V,(b . d)))]}} is non empty set
dom (b . d) is Element of the carrier of (V)
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Source of (V) . (b . d) is Element of the carrier of (V)
[(dom (b . d)),(V,(V,(b . d)))] is V15() Element of [: the carrier of (V),V:]
[: the carrier of (V),V:] is Relation-like non empty set
{(dom (b . d)),(V,(V,(b . d)))} is non empty set
{(dom (b . d))} is non empty set
{{(dom (b . d)),(V,(V,(b . d)))},{(dom (b . d))}} is non empty set
[[(dom (b . d)),(V,(V,(b . d)))],((V,(b . d)) `2)] is V15() set
{[(dom (b . d)),(V,(V,(b . d)))],((V,(b . d)) `2)} is non empty set
{[(dom (b . d)),(V,(V,(b . d)))]} is Relation-like Function-like non empty set
{{[(dom (b . d)),(V,(V,(b . d)))],((V,(b . d)) `2)},{[(dom (b . d)),(V,(V,(b . d)))]}} is non empty set
cod (b . d) is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Target of (V) . (b . d) is Element of the carrier of (V)
[(dom (b . d)),(cod (b . d))] is V15() Element of the carrier of [:(V),(V):]
[:(V),(V):] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of (V), the carrier of (V):] is Relation-like non empty set
[: the carrier' of (V), the carrier' of (V):] is Relation-like non empty set
[: the Source of (V), the Source of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is non empty set
[: the Target of (V), the Target of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
the Comp of (V) is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined the carrier' of (V) -valued Function-like Element of bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):]
[:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is non empty set
|: the Comp of (V), the Comp of (V):| is Relation-like [:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] -defined [: the carrier' of (V), the carrier' of (V):] -valued Function-like Element of bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
[:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is non empty set
CatStr(# [: the carrier of (V), the carrier of (V):],[: the carrier' of (V), the carrier' of (V):],[: the Source of (V), the Source of (V):],[: the Target of (V), the Target of (V):],|: the Comp of (V), the Comp of (V):| #) is non empty non void V59() strict CatStr
the carrier of [:(V),(V):] is non empty set
{(dom (b . d)),(cod (b . d))} is non empty set
{{(dom (b . d)),(cod (b . d))},{(dom (b . d))}} is non empty set
[[(dom (b . d)),(cod (b . d))],((V,(b . d)) `2)] is V15() set
{[(dom (b . d)),(cod (b . d))],((V,(b . d)) `2)} is non empty set
{[(dom (b . d)),(cod (b . d))]} is Relation-like Function-like non empty set
{{[(dom (b . d)),(cod (b . d))],((V,(b . d)) `2)},{[(dom (b . d)),(cod (b . d))]}} is non empty set
Hom ((cod A),a) is Element of bool the carrier' of C
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of C : ( dom b<sub>1</sub> = cod A & cod b<sub>1</sub> = a ) } is set
dom A is Element of the carrier of C
the Source of C . A is Element of the carrier of C
Hom ((dom A),a) is Element of bool the carrier' of C
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of C : ( dom b<sub>1</sub> = dom A & cod b<sub>1</sub> = a ) } is set
[:(Hom ((cod A),a)),(Hom ((dom A),a)):] is Relation-like set
bool [:(Hom ((cod A),a)),(Hom ((dom A),a)):] is non empty set
[(Hom ((cod A),a)),(Hom ((dom A),a))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom ((cod A),a)),(Hom ((dom A),a))} is non empty set
{(Hom ((cod A),a))} is non empty set
{{(Hom ((cod A),a)),(Hom ((dom A),a))},{(Hom ((cod A),a))}} is non empty set
(C,a,A) is Relation-like Hom ((cod A),a) -defined Hom ((dom A),a) -valued Function-like quasi_total Element of bool [:(Hom ((cod A),a)),(Hom ((dom A),a)):]
[[(Hom ((cod A),a)),(Hom ((dom A),a))],(C,a,A)] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod A),a)),(Hom ((dom A),a)):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod A),a)),(Hom ((dom A),a)):]):] is Relation-like non empty set
{[(Hom ((cod A),a)),(Hom ((dom A),a))],(C,a,A)} is non empty set
{[(Hom ((cod A),a)),(Hom ((dom A),a))]} is Relation-like Function-like non empty set
{{[(Hom ((cod A),a)),(Hom ((dom A),a))],(C,a,A)},{[(Hom ((cod A),a)),(Hom ((dom A),a))]}} is non empty set
b . A is Element of the carrier' of (V)
(V,(b . A)) is Element of (V)
(V,(V,(b . A))) is Element of V
(V,(b . A)) `1 is set
((V,(b . A)) `1) `1 is set
(V,(V,(b . A))) is Element of V
((V,(b . A)) `1) `2 is set
[(V,(V,(b . A))),(V,(V,(b . A)))] is V15() Element of [:V,V:]
{(V,(V,(b . A))),(V,(V,(b . A)))} is non empty set
{(V,(V,(b . A)))} is non empty set
{{(V,(V,(b . A))),(V,(V,(b . A)))},{(V,(V,(b . A)))}} is non empty set
(V,(b . A)) `2 is Relation-like Function-like set
[[(V,(V,(b . A))),(V,(V,(b . A)))],((V,(b . A)) `2)] is V15() set
{[(V,(V,(b . A))),(V,(V,(b . A)))],((V,(b . A)) `2)} is non empty set
{[(V,(V,(b . A))),(V,(V,(b . A)))]} is Relation-like Function-like non empty set
{{[(V,(V,(b . A))),(V,(V,(b . A)))],((V,(b . A)) `2)},{[(V,(V,(b . A))),(V,(V,(b . A)))]}} is non empty set
dom (b . A) is Element of the carrier of (V)
the Source of (V) . (b . A) is Element of the carrier of (V)
[(dom (b . A)),(V,(V,(b . A)))] is V15() Element of [: the carrier of (V),V:]
{(dom (b . A)),(V,(V,(b . A)))} is non empty set
{(dom (b . A))} is non empty set
{{(dom (b . A)),(V,(V,(b . A)))},{(dom (b . A))}} is non empty set
[[(dom (b . A)),(V,(V,(b . A)))],((V,(b . A)) `2)] is V15() set
{[(dom (b . A)),(V,(V,(b . A)))],((V,(b . A)) `2)} is non empty set
{[(dom (b . A)),(V,(V,(b . A)))]} is Relation-like Function-like non empty set
{{[(dom (b . A)),(V,(V,(b . A)))],((V,(b . A)) `2)},{[(dom (b . A)),(V,(V,(b . A)))]}} is non empty set
cod (b . A) is Element of the carrier of (V)
the Target of (V) . (b . A) is Element of the carrier of (V)
[(dom (b . A)),(cod (b . A))] is V15() Element of the carrier of [:(V),(V):]
{(dom (b . A)),(cod (b . A))} is non empty set
{{(dom (b . A)),(cod (b . A))},{(dom (b . A))}} is non empty set
[[(dom (b . A)),(cod (b . A))],((V,(b . A)) `2)] is V15() set
{[(dom (b . A)),(cod (b . A))],((V,(b . A)) `2)} is non empty set
{[(dom (b . A)),(cod (b . A))]} is Relation-like Function-like non empty set
{{[(dom (b . A)),(cod (b . A))],((V,(b . A)) `2)},{[(dom (b . A)),(cod (b . A))]}} is non empty set
d (*) A is Element of the carrier' of C
dom (d (*) A) is Element of the carrier of C
the Source of C . (d (*) A) is Element of the carrier of C
cod (d (*) A) is Element of the carrier of C
the Target of C . (d (*) A) is Element of the carrier of C
b . (d (*) A) is Element of the carrier' of (V)
Hom ((cod (d (*) A)),a) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = cod (d (*) A) & cod b₁ = a ) } is set
Hom ((dom (d (*) A)),a) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = dom (d (*) A) & cod b₁ = a ) } is set
[:(Hom ((cod (d (*) A)),a)),(Hom ((dom (d (*) A)),a)):] is Relation-like set
bool [:(Hom ((cod (d (*) A)),a)),(Hom ((dom (d (*) A)),a)):] is non empty set
[(Hom ((cod d),a)),(Hom ((dom A),a))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom ((cod d),a)),(Hom ((dom A),a))} is non empty set
{{(Hom ((cod d),a)),(Hom ((dom A),a))},{(Hom ((cod d),a))}} is non empty set
(C,a,(d (*) A)) is Relation-like Hom ((cod (d (*) A)),a) -defined Hom ((dom (d (*) A)),a) -valued Function-like quasi_total Element of bool [:(Hom ((cod (d (*) A)),a)),(Hom ((dom (d (*) A)),a)):]
[[(Hom ((cod d),a)),(Hom ((dom A),a))],(C,a,(d (*) A))] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod (d (*) A)),a)),(Hom ((dom (d (*) A)),a)):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod (d (*) A)),a)),(Hom ((dom (d (*) A)),a)):]):] is Relation-like non empty set
{[(Hom ((cod d),a)),(Hom ((dom A),a))],(C,a,(d (*) A))} is non empty set
{[(Hom ((cod d),a)),(Hom ((dom A),a))]} is Relation-like Function-like non empty set
{{[(Hom ((cod d),a)),(Hom ((dom A),a))],(C,a,(d (*) A))},{[(Hom ((cod d),a)),(Hom ((dom A),a))]}} is non empty set
[:(Hom ((cod d),a)),(Hom ((dom A),a)):] is Relation-like set
bool [:(Hom ((cod d),a)),(Hom ((dom A),a)):] is non empty set
(C,a,A) * (C,a,d) is Relation-like Hom ((cod d),a) -defined Hom ((dom A),a) -valued Function-like Element of bool [:(Hom ((cod d),a)),(Hom ((dom A),a)):]
[[(Hom ((cod d),a)),(Hom ((dom A),a))],((C,a,A) * (C,a,d))] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod d),a)),(Hom ((dom A),a)):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod d),a)),(Hom ((dom A),a)):]):] is Relation-like non empty set
{[(Hom ((cod d),a)),(Hom ((dom A),a))],((C,a,A) * (C,a,d))} is non empty set
{{[(Hom ((cod d),a)),(Hom ((dom A),a))],((C,a,A) * (C,a,d))},{[(Hom ((cod d),a)),(Hom ((dom A),a))]}} is non empty set
(V,(V,(b . d)),(V,(b . A))) is Element of (V)
(b . A) (*) (b . d) is Element of the carrier' of (V)
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier' of V is non empty set
C is Element of the carrier' of V
dom C is Element of the carrier of V
the carrier of V is non empty set
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Source of V . C is Element of the carrier of V
cod C is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Target of V . C is Element of the carrier of V
a is Element of the carrier' of V
cod a is Element of the carrier of V
the Target of V . a is Element of the carrier of V
Hom ((dom C),(cod a)) is Element of bool the carrier' of V
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom C & cod b₁ = cod a ) } is set
dom a is Element of the carrier of V
the Source of V . a is Element of the carrier of V
Hom ((cod C),(dom a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod C & cod b₁ = dom a ) } is set
Hom ((dom a),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod a ) } is set
Hom ((dom C),(cod C)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom C & cod b₁ = cod C ) } is set
Hom ((dom C),(dom a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom C & cod b₁ = dom a ) } is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier' of V is non empty set
C is Element of the carrier' of V
cod C is Element of the carrier of V
the carrier of V is non empty set
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Target of V . C is Element of the carrier of V
a is Element of the carrier' of V
dom a is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Source of V . a is Element of the carrier of V
Hom ((cod C),(dom a)) is Element of bool the carrier' of V
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod C & cod b₁ = dom a ) } is set
dom C is Element of the carrier of V
the Source of V . C is Element of the carrier of V
cod a is Element of the carrier of V
the Target of V . a is Element of the carrier of V
Hom ((dom C),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom C & cod b₁ = cod a ) } is set
[:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):] is Relation-like set
bool [:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):] is non empty set
d is set
Hom ((dom C),(cod C)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom C & cod b₁ = cod C ) } is set
g9 is Morphism of cod C, dom a
a (*) g9 is Element of the carrier' of V
(a (*) g9) (*) C is Element of the carrier' of V
Hom ((dom a),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod a ) } is set
Hom ((cod C),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod C & cod b₁ = cod a ) } is set
f is Element of the carrier' of V
a (*) f is Element of the carrier' of V
(a (*) f) (*) C is Element of the carrier' of V
d is Relation-like Function-like set
proj1 d is set
proj2 d is set
g9 is Relation-like Hom ((cod C),(dom a)) -defined Hom ((dom C),(cod a)) -valued Function-like quasi_total Element of bool [:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):]
f is Element of the carrier' of V
g9 . f is set
a (*) f is Element of the carrier' of V
(a (*) f) (*) C is Element of the carrier' of V
d is Relation-like Hom ((cod C),(dom a)) -defined Hom ((dom C),(cod a)) -valued Function-like quasi_total Element of bool [:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):]
g9 is Relation-like Hom ((cod C),(dom a)) -defined Hom ((dom C),(cod a)) -valued Function-like quasi_total Element of bool [:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):]
f is set
d . f is set
f9 is Element of the carrier' of V
a (*) f9 is Element of the carrier' of V
(a (*) f9) (*) C is Element of the carrier' of V
g9 . f is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier' of V is non empty set
bool the carrier' of V is non empty set
[:(bool the carrier' of V),(bool the carrier' of V):] is Relation-like non empty set
(V) is non empty set
the carrier of V is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of V : verum } is set
((V)) is non empty set
((V)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (V) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (V) : verum } is set
[:(V),(V):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (V), b₃ is Relation-like Function-like Element of ((V)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
C is Element of the carrier' of V
cod C is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Target of V . C is Element of the carrier of V
dom C is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Source of V . C is Element of the carrier of V
a is Element of the carrier' of V
dom a is Element of the carrier of V
the Source of V . a is Element of the carrier of V
Hom ((cod C),(dom a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod C & cod b₁ = dom a ) } is set
cod a is Element of the carrier of V
the Target of V . a is Element of the carrier of V
Hom ((dom C),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom C & cod b₁ = cod a ) } is set
[:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):] is Relation-like set
bool [:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):] is non empty set
[(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a)))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a)))} is non empty set
{(Hom ((cod C),(dom a)))} is non empty set
{{(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a)))},{(Hom ((cod C),(dom a)))}} is non empty set
(V,C,a) is Relation-like Hom ((cod C),(dom a)) -defined Hom ((dom C),(cod a)) -valued Function-like quasi_total Element of bool [:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):]
[[(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a)))],(V,C,a)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):]):] is Relation-like non empty set
{[(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a)))],(V,C,a)} is non empty set
{[(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a)))]} is Relation-like Function-like non empty set
{{[(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a)))],(V,C,a)},{[(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a)))]}} is non empty set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
the carrier' of V is non empty set
C is Element of the carrier of V
id C is Morphism of C,C
a is Element of the carrier' of V
(V,(id C),a) is Relation-like Hom ((cod (id C)),(dom a)) -defined Hom ((dom (id C)),(cod a)) -valued Function-like quasi_total Element of bool [:(Hom ((cod (id C)),(dom a))),(Hom ((dom (id C)),(cod a))):]
cod (id C) is Element of the carrier of V
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Target of V . (id C) is Element of the carrier of V
dom a is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Source of V . a is Element of the carrier of V
Hom ((cod (id C)),(dom a)) is Element of bool the carrier' of V
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod (id C) & cod b₁ = dom a ) } is set
dom (id C) is Element of the carrier of V
the Source of V . (id C) is Element of the carrier of V
cod a is Element of the carrier of V
the Target of V . a is Element of the carrier of V
Hom ((dom (id C)),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom (id C) & cod b₁ = cod a ) } is set
[:(Hom ((cod (id C)),(dom a))),(Hom ((dom (id C)),(cod a))):] is Relation-like set
bool [:(Hom ((cod (id C)),(dom a))),(Hom ((dom (id C)),(cod a))):] is non empty set
(V,C,a) is Relation-like Hom (C,(dom a)) -defined Hom (C,(cod a)) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom a))),(Hom (C,(cod a))):]
Hom (C,(dom a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom a ) } is set
Hom (C,(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod a ) } is set
[:(Hom (C,(dom a))),(Hom (C,(cod a))):] is Relation-like set
bool [:(Hom (C,(dom a))),(Hom (C,(cod a))):] is non empty set
(V,a,(id C)) is Relation-like Hom ((cod a),(dom (id C))) -defined Hom ((dom a),(cod (id C))) -valued Function-like quasi_total Element of bool [:(Hom ((cod a),(dom (id C)))),(Hom ((dom a),(cod (id C)))):]
Hom ((cod a),(dom (id C))) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod a & cod b₁ = dom (id C) ) } is set
Hom ((dom a),(cod (id C))) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod (id C) ) } is set
[:(Hom ((cod a),(dom (id C)))),(Hom ((dom a),(cod (id C)))):] is Relation-like set
bool [:(Hom ((cod a),(dom (id C)))),(Hom ((dom a),(cod (id C)))):] is non empty set
(V,C,a) is Relation-like Hom ((cod a),C) -defined Hom ((dom a),C) -valued Function-like quasi_total Element of bool [:(Hom ((cod a),C)),(Hom ((dom a),C)):]
Hom ((cod a),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod a & cod b₁ = C ) } is set
Hom ((dom a),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = C ) } is set
[:(Hom ((cod a),C)),(Hom ((dom a),C)):] is Relation-like set
bool [:(Hom ((cod a),C)),(Hom ((dom a),C)):] is non empty set
Hom ((dom a),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod a ) } is set
proj1 (V,(id C),a) is set
proj1 (V,C,a) is set
b is set
A is Element of the carrier' of V
dom A is Element of the carrier of V
the Source of V . A is Element of the carrier of V
cod A is Element of the carrier of V
the Target of V . A is Element of the carrier of V
(V,(id C),a) . b is set
a (*) A is Element of the carrier' of V
(a (*) A) (*) (id C) is Element of the carrier' of V
A (*) (id C) is Element of the carrier' of V
a (*) (A (*) (id C)) is Element of the carrier' of V
(V,C,a) . b is set
proj1 (V,a,(id C)) is set
proj1 (V,C,a) is set
b is set
A is Element of the carrier' of V
cod A is Element of the carrier of V
the Target of V . A is Element of the carrier of V
(V,a,(id C)) . b is set
(id C) (*) A is Element of the carrier' of V
((id C) (*) A) (*) a is Element of the carrier' of V
A (*) a is Element of the carrier' of V
(V,C,a) . b is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
C is Element of the carrier of V
id C is Morphism of C,C
a is Element of the carrier of V
id a is Morphism of a,a
(V,(id C),(id a)) is Relation-like Hom ((cod (id C)),(dom (id a))) -defined Hom ((dom (id C)),(cod (id a))) -valued Function-like quasi_total Element of bool [:(Hom ((cod (id C)),(dom (id a)))),(Hom ((dom (id C)),(cod (id a)))):]
cod (id C) is Element of the carrier of V
the carrier' of V is non empty set
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Target of V . (id C) is Element of the carrier of V
dom (id a) is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Source of V . (id a) is Element of the carrier of V
Hom ((cod (id C)),(dom (id a))) is Element of bool the carrier' of V
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod (id C) & cod b₁ = dom (id a) ) } is set
dom (id C) is Element of the carrier of V
the Source of V . (id C) is Element of the carrier of V
cod (id a) is Element of the carrier of V
the Target of V . (id a) is Element of the carrier of V
Hom ((dom (id C)),(cod (id a))) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom (id C) & cod b₁ = cod (id a) ) } is set
[:(Hom ((cod (id C)),(dom (id a)))),(Hom ((dom (id C)),(cod (id a)))):] is Relation-like set
bool [:(Hom ((cod (id C)),(dom (id a)))),(Hom ((dom (id C)),(cod (id a)))):] is non empty set
Hom (C,a) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = a ) } is set
id (Hom (C,a)) is Relation-like Hom (C,a) -defined Hom (C,a) -valued Function-like one-to-one total quasi_total Element of bool [:(Hom (C,a)),(Hom (C,a)):]
[:(Hom (C,a)),(Hom (C,a)):] is Relation-like set
bool [:(Hom (C,a)),(Hom (C,a)):] is non empty set
(V,C,(id a)) is Relation-like Hom (C,(dom (id a))) -defined Hom (C,(cod (id a))) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom (id a)))),(Hom (C,(cod (id a)))):]
Hom (C,(dom (id a))) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom (id a) ) } is set
Hom (C,(cod (id a))) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod (id a) ) } is set
[:(Hom (C,(dom (id a)))),(Hom (C,(cod (id a)))):] is Relation-like set
bool [:(Hom (C,(dom (id a)))),(Hom (C,(cod (id a)))):] is non empty set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier' of V is non empty set
C is Element of the carrier' of V
cod C is Element of the carrier of V
the carrier of V is non empty set
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Target of V . C is Element of the carrier of V
dom C is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Source of V . C is Element of the carrier of V
a is Element of the carrier' of V
(V,C,a) is Relation-like Hom ((cod C),(dom a)) -defined Hom ((dom C),(cod a)) -valued Function-like quasi_total Element of bool [:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):]
dom a is Element of the carrier of V
the Source of V . a is Element of the carrier of V
Hom ((cod C),(dom a)) is Element of bool the carrier' of V
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod C & cod b₁ = dom a ) } is set
cod a is Element of the carrier of V
the Target of V . a is Element of the carrier of V
Hom ((dom C),(cod a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom C & cod b₁ = cod a ) } is set
[:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):] is Relation-like set
bool [:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):] is non empty set
Hom ((dom C),(dom a)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom C & cod b₁ = dom a ) } is set
(V,(dom a),C) is Relation-like Hom ((cod C),(dom a)) -defined Hom ((dom C),(dom a)) -valued Function-like quasi_total Element of bool [:(Hom ((cod C),(dom a))),(Hom ((dom C),(dom a))):]
[:(Hom ((cod C),(dom a))),(Hom ((dom C),(dom a))):] is Relation-like set
bool [:(Hom ((cod C),(dom a))),(Hom ((dom C),(dom a))):] is non empty set
(V,(dom C),a) is Relation-like Hom ((dom C),(dom a)) -defined Hom ((dom C),(cod a)) -valued Function-like quasi_total Element of bool [:(Hom ((dom C),(dom a))),(Hom ((dom C),(cod a))):]
[:(Hom ((dom C),(dom a))),(Hom ((dom C),(cod a))):] is Relation-like set
bool [:(Hom ((dom C),(dom a))),(Hom ((dom C),(cod a))):] is non empty set
(V,(dom C),a) * (V,(dom a),C) is Relation-like Hom ((cod C),(dom a)) -defined Hom ((dom C),(cod a)) -valued Function-like Element of bool [:(Hom ((cod C),(dom a))),(Hom ((dom C),(cod a))):]
proj1 (V,C,a) is set
Hom ((dom C),(cod C)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom C & cod b₁ = cod C ) } is set
proj1 ((V,(dom C),a) * (V,(dom a),C)) is set
b is set
A is Element of the carrier' of V
dom A is Element of the carrier of V
the Source of V . A is Element of the carrier of V
A (*) C is Element of the carrier' of V
dom (A (*) C) is Element of the carrier of V
the Source of V . (A (*) C) is Element of the carrier of V
cod A is Element of the carrier of V
the Target of V . A is Element of the carrier of V
cod (A (*) C) is Element of the carrier of V
the Target of V . (A (*) C) is Element of the carrier of V
(V,C,a) . b is set
a (*) A is Element of the carrier' of V
(a (*) A) (*) C is Element of the carrier' of V
a (*) (A (*) C) is Element of the carrier' of V
(V,(dom C),a) . (A (*) C) is set
(V,(dom a),C) . A is set
(V,(dom C),a) . ((V,(dom a),C) . A) is set
((V,(dom C),a) * (V,(dom a),C)) . b is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier' of V is non empty set
C is Element of the carrier' of V
cod C is Element of the carrier of V
the carrier of V is non empty set
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
the Target of V . C is Element of the carrier of V
dom C is Element of the carrier of V
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
the Source of V . C is Element of the carrier of V
a is Element of the carrier' of V
dom a is Element of the carrier of V
the Source of V . a is Element of the carrier of V
a (*) C is Element of the carrier' of V
cod a is Element of the carrier of V
the Target of V . a is Element of the carrier of V
b is Element of the carrier' of V
dom b is Element of the carrier of V
the Source of V . b is Element of the carrier of V
Hom ((cod C),(dom b)) is Element of bool the carrier' of V
bool the carrier' of V is non empty set
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod C & cod b₁ = dom b ) } is set
cod b is Element of the carrier of V
the Target of V . b is Element of the carrier of V
Hom ((dom C),(cod b)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom C & cod b₁ = cod b ) } is set
(V,C,b) is Relation-like Hom ((cod C),(dom b)) -defined Hom ((dom C),(cod b)) -valued Function-like quasi_total Element of bool [:(Hom ((cod C),(dom b))),(Hom ((dom C),(cod b))):]
[:(Hom ((cod C),(dom b))),(Hom ((dom C),(cod b))):] is Relation-like set
bool [:(Hom ((cod C),(dom b))),(Hom ((dom C),(cod b))):] is non empty set
A is Element of the carrier' of V
cod A is Element of the carrier of V
the Target of V . A is Element of the carrier of V
b (*) A is Element of the carrier' of V
(V,(a (*) C),(b (*) A)) is Relation-like Hom ((cod (a (*) C)),(dom (b (*) A))) -defined Hom ((dom (a (*) C)),(cod (b (*) A))) -valued Function-like quasi_total Element of bool [:(Hom ((cod (a (*) C)),(dom (b (*) A)))),(Hom ((dom (a (*) C)),(cod (b (*) A)))):]
cod (a (*) C) is Element of the carrier of V
the Target of V . (a (*) C) is Element of the carrier of V
dom (b (*) A) is Element of the carrier of V
the Source of V . (b (*) A) is Element of the carrier of V
Hom ((cod (a (*) C)),(dom (b (*) A))) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod (a (*) C) & cod b₁ = dom (b (*) A) ) } is set
dom (a (*) C) is Element of the carrier of V
the Source of V . (a (*) C) is Element of the carrier of V
cod (b (*) A) is Element of the carrier of V
the Target of V . (b (*) A) is Element of the carrier of V
Hom ((dom (a (*) C)),(cod (b (*) A))) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom (a (*) C) & cod b₁ = cod (b (*) A) ) } is set
[:(Hom ((cod (a (*) C)),(dom (b (*) A)))),(Hom ((dom (a (*) C)),(cod (b (*) A)))):] is Relation-like set
bool [:(Hom ((cod (a (*) C)),(dom (b (*) A)))),(Hom ((dom (a (*) C)),(cod (b (*) A)))):] is non empty set
dom A is Element of the carrier of V
the Source of V . A is Element of the carrier of V
Hom ((cod a),(dom A)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod a & cod b₁ = dom A ) } is set
Hom ((dom a),(cod A)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom a & cod b₁ = cod A ) } is set
(V,a,A) is Relation-like Hom ((cod a),(dom A)) -defined Hom ((dom a),(cod A)) -valued Function-like quasi_total Element of bool [:(Hom ((cod a),(dom A))),(Hom ((dom a),(cod A))):]
[:(Hom ((cod a),(dom A))),(Hom ((dom a),(cod A))):] is Relation-like set
bool [:(Hom ((cod a),(dom A))),(Hom ((dom a),(cod A))):] is non empty set
(V,C,b) * (V,a,A) is Relation-like Hom ((cod a),(dom A)) -defined Hom ((dom C),(cod b)) -valued Function-like Element of bool [:(Hom ((cod a),(dom A))),(Hom ((dom C),(cod b))):]
[:(Hom ((cod a),(dom A))),(Hom ((dom C),(cod b))):] is Relation-like set
bool [:(Hom ((cod a),(dom A))),(Hom ((dom C),(cod b))):] is non empty set
proj1 (V,(a (*) C),(b (*) A)) is set
proj1 ((V,C,b) * (V,a,A)) is set
f9 is set
(V,a,A) . f9 is set
k is Element of the carrier' of V
dom k is Element of the carrier of V
the Source of V . k is Element of the carrier of V
k (*) (a (*) C) is Element of the carrier' of V
cod (k (*) (a (*) C)) is Element of the carrier of V
the Target of V . (k (*) (a (*) C)) is Element of the carrier of V
cod k is Element of the carrier of V
the Target of V . k is Element of the carrier of V
A (*) k is Element of the carrier' of V
dom (A (*) k) is Element of the carrier of V
the Source of V . (A (*) k) is Element of the carrier of V
(A (*) k) (*) a is Element of the carrier' of V
dom ((A (*) k) (*) a) is Element of the carrier of V
the Source of V . ((A (*) k) (*) a) is Element of the carrier of V
cod (A (*) k) is Element of the carrier of V
the Target of V . (A (*) k) is Element of the carrier of V
cod ((A (*) k) (*) a) is Element of the carrier of V
the Target of V . ((A (*) k) (*) a) is Element of the carrier of V
(V,(a (*) C),(b (*) A)) . f9 is set
(b (*) A) (*) k is Element of the carrier' of V
((b (*) A) (*) k) (*) (a (*) C) is Element of the carrier' of V
(b (*) A) (*) (k (*) (a (*) C)) is Element of the carrier' of V
A (*) (k (*) (a (*) C)) is Element of the carrier' of V
b (*) (A (*) (k (*) (a (*) C))) is Element of the carrier' of V
(A (*) k) (*) (a (*) C) is Element of the carrier' of V
b (*) ((A (*) k) (*) (a (*) C)) is Element of the carrier' of V
((A (*) k) (*) a) (*) C is Element of the carrier' of V
b (*) (((A (*) k) (*) a) (*) C) is Element of the carrier' of V
b (*) ((A (*) k) (*) a) is Element of the carrier' of V
(b (*) ((A (*) k) (*) a)) (*) C is Element of the carrier' of V
l is Element of the carrier' of V
b (*) l is Element of the carrier' of V
(b (*) l) (*) C is Element of the carrier' of V
(V,C,b) . l is set
((V,C,b) * (V,a,A)) . f9 is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[:V,V:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
[: the carrier of V, the carrier of V:] is Relation-like non empty set
the carrier' of V is non empty set
[: the carrier' of V, the carrier' of V:] is Relation-like non empty set
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
[: the Source of V, the Source of V:] is Relation-like [: the carrier' of V, the carrier' of V:] -defined [: the carrier of V, the carrier of V:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of V, the carrier' of V:],[: the carrier of V, the carrier of V:]:]
[:[: the carrier' of V, the carrier' of V:],[: the carrier of V, the carrier of V:]:] is Relation-like non empty set
bool [:[: the carrier' of V, the carrier' of V:],[: the carrier of V, the carrier of V:]:] is non empty set
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the Target of V, the Target of V:] is Relation-like [: the carrier' of V, the carrier' of V:] -defined [: the carrier of V, the carrier of V:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of V, the carrier' of V:],[: the carrier of V, the carrier of V:]:]
the Comp of V is Relation-like [: the carrier' of V, the carrier' of V:] -defined the carrier' of V -valued Function-like Element of bool [:[: the carrier' of V, the carrier' of V:], the carrier' of V:]
[:[: the carrier' of V, the carrier' of V:], the carrier' of V:] is Relation-like non empty set
bool [:[: the carrier' of V, the carrier' of V:], the carrier' of V:] is non empty set
|: the Comp of V, the Comp of V:| is Relation-like [:[: the carrier' of V, the carrier' of V:],[: the carrier' of V, the carrier' of V:]:] -defined [: the carrier' of V, the carrier' of V:] -valued Function-like Element of bool [:[:[: the carrier' of V, the carrier' of V:],[: the carrier' of V, the carrier' of V:]:],[: the carrier' of V, the carrier' of V:]:]
[:[: the carrier' of V, the carrier' of V:],[: the carrier' of V, the carrier' of V:]:] is Relation-like non empty set
[:[:[: the carrier' of V, the carrier' of V:],[: the carrier' of V, the carrier' of V:]:],[: the carrier' of V, the carrier' of V:]:] is Relation-like non empty set
bool [:[:[: the carrier' of V, the carrier' of V:],[: the carrier' of V, the carrier' of V:]:],[: the carrier' of V, the carrier' of V:]:] is non empty set
CatStr(# [: the carrier of V, the carrier of V:],[: the carrier' of V, the carrier' of V:],[: the Source of V, the Source of V:],[: the Target of V, the Target of V:],|: the Comp of V, the Comp of V:| #) is non empty non void V59() strict CatStr
the carrier' of [:V,V:] is non empty set
(V) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of V : verum } is set
((V)) is non empty set
((V)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (V) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (V) : verum } is set
[:(V),(V):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (V), b₃ is Relation-like Function-like Element of ((V)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
[: the carrier' of [:V,V:],((V)):] is Relation-like non empty set
bool [: the carrier' of [:V,V:],((V)):] is non empty set
bool the carrier' of V is non empty set
[:(bool the carrier' of V),(bool the carrier' of V):] is Relation-like non empty set
b is set
A is Element of the carrier' of V
d is Element of the carrier' of V
[A,d] is V15() Element of the carrier' of [:V,V:]
{A,d} is non empty set
{A} is non empty set
{{A,d},{A}} is non empty set
cod A is Element of the carrier of V
the Target of V . A is Element of the carrier of V
dom d is Element of the carrier of V
the Source of V . d is Element of the carrier of V
Hom ((cod A),(dom d)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod A & cod b₁ = dom d ) } is set
dom A is Element of the carrier of V
the Source of V . A is Element of the carrier of V
cod d is Element of the carrier of V
the Target of V . d is Element of the carrier of V
Hom ((dom A),(cod d)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom A & cod b₁ = cod d ) } is set
[:(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d))):] is Relation-like set
bool [:(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d))):] is non empty set
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d))):]):] is Relation-like non empty set
[(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d)))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d)))} is non empty set
{(Hom ((cod A),(dom d)))} is non empty set
{{(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d)))},{(Hom ((cod A),(dom d)))}} is non empty set
(V,A,d) is Relation-like Hom ((cod A),(dom d)) -defined Hom ((dom A),(cod d)) -valued Function-like quasi_total Element of bool [:(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d))):]
[[(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d)))],(V,A,d)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d))):]):]
{[(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d)))],(V,A,d)} is non empty set
{[(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d)))]} is Relation-like Function-like non empty set
{{[(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d)))],(V,A,d)},{[(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d)))]}} is non empty set
g9 is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod A),(dom d))),(Hom ((dom A),(cod d))):]):]
f is Element of the carrier' of V
cod f is Element of the carrier of V
the Target of V . f is Element of the carrier of V
dom f is Element of the carrier of V
the Source of V . f is Element of the carrier of V
f9 is Element of the carrier' of V
[f,f9] is V15() Element of the carrier' of [:V,V:]
{f,f9} is non empty set
{f} is non empty set
{{f,f9},{f}} is non empty set
dom f9 is Element of the carrier of V
the Source of V . f9 is Element of the carrier of V
Hom ((cod f),(dom f9)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod f & cod b₁ = dom f9 ) } is set
cod f9 is Element of the carrier of V
the Target of V . f9 is Element of the carrier of V
Hom ((dom f),(cod f9)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom f & cod b₁ = cod f9 ) } is set
[:(Hom ((cod f),(dom f9))),(Hom ((dom f),(cod f9))):] is Relation-like set
bool [:(Hom ((cod f),(dom f9))),(Hom ((dom f),(cod f9))):] is non empty set
[(Hom ((cod f),(dom f9))),(Hom ((dom f),(cod f9)))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod f),(dom f9))),(Hom ((dom f),(cod f9)))} is non empty set
{(Hom ((cod f),(dom f9)))} is non empty set
{{(Hom ((cod f),(dom f9))),(Hom ((dom f),(cod f9)))},{(Hom ((cod f),(dom f9)))}} is non empty set
(V,f,f9) is Relation-like Hom ((cod f),(dom f9)) -defined Hom ((dom f),(cod f9)) -valued Function-like quasi_total Element of bool [:(Hom ((cod f),(dom f9))),(Hom ((dom f),(cod f9))):]
[[(Hom ((cod f),(dom f9))),(Hom ((dom f),(cod f9)))],(V,f,f9)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod f),(dom f9))),(Hom ((dom f),(cod f9))):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod f),(dom f9))),(Hom ((dom f),(cod f9))):]):] is Relation-like non empty set
{[(Hom ((cod f),(dom f9))),(Hom ((dom f),(cod f9)))],(V,f,f9)} is non empty set
{[(Hom ((cod f),(dom f9))),(Hom ((dom f),(cod f9)))]} is Relation-like Function-like non empty set
{{[(Hom ((cod f),(dom f9))),(Hom ((dom f),(cod f9)))],(V,f,f9)},{[(Hom ((cod f),(dom f9))),(Hom ((dom f),(cod f9)))]}} is non empty set
b is Relation-like Function-like set
proj1 b is set
proj2 b is set
A is Relation-like the carrier' of [:V,V:] -defined ((V)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:V,V:],((V)):]
d is Element of the carrier' of V
g9 is Element of the carrier' of V
[d,g9] is V15() Element of the carrier' of [:V,V:]
{d,g9} is non empty set
{d} is non empty set
{{d,g9},{d}} is non empty set
A . [d,g9] is Element of ((V))
cod d is Element of the carrier of V
the Target of V . d is Element of the carrier of V
dom g9 is Element of the carrier of V
the Source of V . g9 is Element of the carrier of V
Hom ((cod d),(dom g9)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod d & cod b₁ = dom g9 ) } is set
dom d is Element of the carrier of V
the Source of V . d is Element of the carrier of V
cod g9 is Element of the carrier of V
the Target of V . g9 is Element of the carrier of V
Hom ((dom d),(cod g9)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom d & cod b₁ = cod g9 ) } is set
[:(Hom ((cod d),(dom g9))),(Hom ((dom d),(cod g9))):] is Relation-like set
bool [:(Hom ((cod d),(dom g9))),(Hom ((dom d),(cod g9))):] is non empty set
[(Hom ((cod d),(dom g9))),(Hom ((dom d),(cod g9)))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod d),(dom g9))),(Hom ((dom d),(cod g9)))} is non empty set
{(Hom ((cod d),(dom g9)))} is non empty set
{{(Hom ((cod d),(dom g9))),(Hom ((dom d),(cod g9)))},{(Hom ((cod d),(dom g9)))}} is non empty set
(V,d,g9) is Relation-like Hom ((cod d),(dom g9)) -defined Hom ((dom d),(cod g9)) -valued Function-like quasi_total Element of bool [:(Hom ((cod d),(dom g9))),(Hom ((dom d),(cod g9))):]
[[(Hom ((cod d),(dom g9))),(Hom ((dom d),(cod g9)))],(V,d,g9)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod d),(dom g9))),(Hom ((dom d),(cod g9))):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod d),(dom g9))),(Hom ((dom d),(cod g9))):]):] is Relation-like non empty set
{[(Hom ((cod d),(dom g9))),(Hom ((dom d),(cod g9)))],(V,d,g9)} is non empty set
{[(Hom ((cod d),(dom g9))),(Hom ((dom d),(cod g9)))]} is Relation-like Function-like non empty set
{{[(Hom ((cod d),(dom g9))),(Hom ((dom d),(cod g9)))],(V,d,g9)},{[(Hom ((cod d),(dom g9))),(Hom ((dom d),(cod g9)))]}} is non empty set
C is Relation-like the carrier' of [:V,V:] -defined ((V)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:V,V:],((V)):]
a is Relation-like the carrier' of [:V,V:] -defined ((V)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:V,V:],((V)):]
b is Element of the carrier' of V
A is Element of the carrier' of V
C . (b,A) is set
[b,A] is V15() set
{b,A} is non empty set
{b} is non empty set
{{b,A},{b}} is non empty set
C . [b,A] is set
cod b is Element of the carrier of V
the Target of V . b is Element of the carrier of V
dom A is Element of the carrier of V
the Source of V . A is Element of the carrier of V
Hom ((cod b),(dom A)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod b & cod b₁ = dom A ) } is set
dom b is Element of the carrier of V
the Source of V . b is Element of the carrier of V
cod A is Element of the carrier of V
the Target of V . A is Element of the carrier of V
Hom ((dom b),(cod A)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom b & cod b₁ = cod A ) } is set
[:(Hom ((cod b),(dom A))),(Hom ((dom b),(cod A))):] is Relation-like set
bool [:(Hom ((cod b),(dom A))),(Hom ((dom b),(cod A))):] is non empty set
[(Hom ((cod b),(dom A))),(Hom ((dom b),(cod A)))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod b),(dom A))),(Hom ((dom b),(cod A)))} is non empty set
{(Hom ((cod b),(dom A)))} is non empty set
{{(Hom ((cod b),(dom A))),(Hom ((dom b),(cod A)))},{(Hom ((cod b),(dom A)))}} is non empty set
(V,b,A) is Relation-like Hom ((cod b),(dom A)) -defined Hom ((dom b),(cod A)) -valued Function-like quasi_total Element of bool [:(Hom ((cod b),(dom A))),(Hom ((dom b),(cod A))):]
[[(Hom ((cod b),(dom A))),(Hom ((dom b),(cod A)))],(V,b,A)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod b),(dom A))),(Hom ((dom b),(cod A))):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod b),(dom A))),(Hom ((dom b),(cod A))):]):] is Relation-like non empty set
{[(Hom ((cod b),(dom A))),(Hom ((dom b),(cod A)))],(V,b,A)} is non empty set
{[(Hom ((cod b),(dom A))),(Hom ((dom b),(cod A)))]} is Relation-like Function-like non empty set
{{[(Hom ((cod b),(dom A))),(Hom ((dom b),(cod A)))],(V,b,A)},{[(Hom ((cod b),(dom A))),(Hom ((dom b),(cod A)))]}} is non empty set
a . (b,A) is set
a . [b,A] is set
V is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of V is non empty set
(V) is Relation-like the carrier' of [:V,V:] -defined ((V)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:V,V:],((V)):]
[:V,V:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of V, the carrier of V:] is Relation-like non empty set
the carrier' of V is non empty set
[: the carrier' of V, the carrier' of V:] is Relation-like non empty set
the Source of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the carrier' of V, the carrier of V:] is Relation-like non empty set
bool [: the carrier' of V, the carrier of V:] is non empty set
[: the Source of V, the Source of V:] is Relation-like [: the carrier' of V, the carrier' of V:] -defined [: the carrier of V, the carrier of V:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of V, the carrier' of V:],[: the carrier of V, the carrier of V:]:]
[:[: the carrier' of V, the carrier' of V:],[: the carrier of V, the carrier of V:]:] is Relation-like non empty set
bool [:[: the carrier' of V, the carrier' of V:],[: the carrier of V, the carrier of V:]:] is non empty set
the Target of V is Relation-like the carrier' of V -defined the carrier of V -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V, the carrier of V:]
[: the Target of V, the Target of V:] is Relation-like [: the carrier' of V, the carrier' of V:] -defined [: the carrier of V, the carrier of V:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of V, the carrier' of V:],[: the carrier of V, the carrier of V:]:]
the Comp of V is Relation-like [: the carrier' of V, the carrier' of V:] -defined the carrier' of V -valued Function-like Element of bool [:[: the carrier' of V, the carrier' of V:], the carrier' of V:]
[:[: the carrier' of V, the carrier' of V:], the carrier' of V:] is Relation-like non empty set
bool [:[: the carrier' of V, the carrier' of V:], the carrier' of V:] is non empty set
|: the Comp of V, the Comp of V:| is Relation-like [:[: the carrier' of V, the carrier' of V:],[: the carrier' of V, the carrier' of V:]:] -defined [: the carrier' of V, the carrier' of V:] -valued Function-like Element of bool [:[:[: the carrier' of V, the carrier' of V:],[: the carrier' of V, the carrier' of V:]:],[: the carrier' of V, the carrier' of V:]:]
[:[: the carrier' of V, the carrier' of V:],[: the carrier' of V, the carrier' of V:]:] is Relation-like non empty set
[:[:[: the carrier' of V, the carrier' of V:],[: the carrier' of V, the carrier' of V:]:],[: the carrier' of V, the carrier' of V:]:] is Relation-like non empty set
bool [:[:[: the carrier' of V, the carrier' of V:],[: the carrier' of V, the carrier' of V:]:],[: the carrier' of V, the carrier' of V:]:] is non empty set
CatStr(# [: the carrier of V, the carrier of V:],[: the carrier' of V, the carrier' of V:],[: the Source of V, the Source of V:],[: the Target of V, the Target of V:],|: the Comp of V, the Comp of V:| #) is non empty non void V59() strict CatStr
the carrier' of [:V,V:] is non empty set
(V) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of V : verum } is set
((V)) is non empty set
((V)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (V) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (V) : verum } is set
[:(V),(V):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (V), b₃ is Relation-like Function-like Element of ((V)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
[: the carrier' of [:V,V:],((V)):] is Relation-like non empty set
bool [: the carrier' of [:V,V:],((V)):] is non empty set
curry (V) is Relation-like Function-like set
curry' (V) is Relation-like Function-like set
C is Element of the carrier of V
(V,C) is Relation-like the carrier' of V -defined ((V)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V,((V)):]
[: the carrier' of V,((V)):] is Relation-like non empty set
bool [: the carrier' of V,((V)):] is non empty set
id C is Morphism of C,C
(curry (V)) . (id C) is set
(V,C) is Relation-like the carrier' of V -defined ((V)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V,((V)):]
(curry' (V)) . (id C) is set
[:[: the carrier' of V, the carrier' of V:],((V)):] is Relation-like non empty set
bool [:[: the carrier' of V, the carrier' of V:],((V)):] is non empty set
Funcs ( the carrier' of V,((V))) is functional non empty FUNCTION_DOMAIN of the carrier' of V,((V))
a is Relation-like [: the carrier' of V, the carrier' of V:] -defined ((V)) -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of V, the carrier' of V:],((V)):]
curry a is Relation-like the carrier' of V -defined Funcs ( the carrier' of V,((V))) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V,(Funcs ( the carrier' of V,((V)))):]
[: the carrier' of V,(Funcs ( the carrier' of V,((V)))):] is Relation-like non empty set
bool [: the carrier' of V,(Funcs ( the carrier' of V,((V)))):] is non empty set
(curry a) . (id C) is Relation-like the carrier' of V -defined ((V)) -valued Function-like total quasi_total Element of Funcs ( the carrier' of V,((V)))
b is Element of the carrier' of V
((curry a) . (id C)) . b is Element of ((V))
a . ((id C),b) is Element of ((V))
[(id C),b] is V15() set
{(id C),b} is non empty set
{(id C)} is non empty set
{{(id C),b},{(id C)}} is non empty set
a . [(id C),b] is set
bool the carrier' of V is non empty set
[:(bool the carrier' of V),(bool the carrier' of V):] is Relation-like non empty set
cod (id C) is Element of the carrier of V
the Target of V . (id C) is Element of the carrier of V
dom b is Element of the carrier of V
the Source of V . b is Element of the carrier of V
Hom ((cod (id C)),(dom b)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod (id C) & cod b₁ = dom b ) } is set
dom (id C) is Element of the carrier of V
the Source of V . (id C) is Element of the carrier of V
cod b is Element of the carrier of V
the Target of V . b is Element of the carrier of V
Hom ((dom (id C)),(cod b)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom (id C) & cod b₁ = cod b ) } is set
[:(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b))):] is Relation-like set
bool [:(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b))):] is non empty set
[(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b)))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b)))} is non empty set
{(Hom ((cod (id C)),(dom b)))} is non empty set
{{(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b)))},{(Hom ((cod (id C)),(dom b)))}} is non empty set
(V,(id C),b) is Relation-like Hom ((cod (id C)),(dom b)) -defined Hom ((dom (id C)),(cod b)) -valued Function-like quasi_total Element of bool [:(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b))):]
[[(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b)))],(V,(id C),b)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b))):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b))):]):] is Relation-like non empty set
{[(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b)))],(V,(id C),b)} is non empty set
{[(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b)))]} is Relation-like Function-like non empty set
{{[(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b)))],(V,(id C),b)},{[(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b)))]}} is non empty set
Hom (C,(dom b)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = dom b ) } is set
Hom (C,(cod b)) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = C & cod b₁ = cod b ) } is set
[:(Hom (C,(dom b))),(Hom (C,(cod b))):] is Relation-like set
bool [:(Hom (C,(dom b))),(Hom (C,(cod b))):] is non empty set
(V,C,b) is Relation-like Hom (C,(dom b)) -defined Hom (C,(cod b)) -valued Function-like quasi_total Element of bool [:(Hom (C,(dom b))),(Hom (C,(cod b))):]
[[(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b)))],(V,C,b)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom b))),(Hom (C,(cod b))):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom b))),(Hom (C,(cod b))):]):] is Relation-like non empty set
{[(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b)))],(V,C,b)} is non empty set
{{[(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b)))],(V,C,b)},{[(Hom ((cod (id C)),(dom b))),(Hom ((dom (id C)),(cod b)))]}} is non empty set
[(Hom (C,(dom b))),(Hom ((dom (id C)),(cod b)))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom (C,(dom b))),(Hom ((dom (id C)),(cod b)))} is non empty set
{(Hom (C,(dom b)))} is non empty set
{{(Hom (C,(dom b))),(Hom ((dom (id C)),(cod b)))},{(Hom (C,(dom b)))}} is non empty set
[[(Hom (C,(dom b))),(Hom ((dom (id C)),(cod b)))],(V,C,b)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom b))),(Hom (C,(cod b))):]):]
{[(Hom (C,(dom b))),(Hom ((dom (id C)),(cod b)))],(V,C,b)} is non empty set
{[(Hom (C,(dom b))),(Hom ((dom (id C)),(cod b)))]} is Relation-like Function-like non empty set
{{[(Hom (C,(dom b))),(Hom ((dom (id C)),(cod b)))],(V,C,b)},{[(Hom (C,(dom b))),(Hom ((dom (id C)),(cod b)))]}} is non empty set
[(Hom (C,(dom b))),(Hom (C,(cod b)))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom (C,(dom b))),(Hom (C,(cod b)))} is non empty set
{{(Hom (C,(dom b))),(Hom (C,(cod b)))},{(Hom (C,(dom b)))}} is non empty set
[[(Hom (C,(dom b))),(Hom (C,(cod b)))],(V,C,b)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom (C,(dom b))),(Hom (C,(cod b))):]):]
{[(Hom (C,(dom b))),(Hom (C,(cod b)))],(V,C,b)} is non empty set
{[(Hom (C,(dom b))),(Hom (C,(cod b)))]} is Relation-like Function-like non empty set
{{[(Hom (C,(dom b))),(Hom (C,(cod b)))],(V,C,b)},{[(Hom (C,(dom b))),(Hom (C,(cod b)))]}} is non empty set
(V,C) . b is Element of ((V))
curry' a is Relation-like the carrier' of V -defined Funcs ( the carrier' of V,((V))) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of V,(Funcs ( the carrier' of V,((V)))):]
(curry' a) . (id C) is Relation-like the carrier' of V -defined ((V)) -valued Function-like total quasi_total Element of Funcs ( the carrier' of V,((V)))
b is Element of the carrier' of V
((curry' a) . (id C)) . b is Element of ((V))
a . (b,(id C)) is Element of ((V))
[b,(id C)] is V15() set
{b,(id C)} is non empty set
{b} is non empty set
{{b,(id C)},{b}} is non empty set
a . [b,(id C)] is set
cod b is Element of the carrier of V
the Target of V . b is Element of the carrier of V
Hom ((cod b),(dom (id C))) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod b & cod b₁ = dom (id C) ) } is set
dom b is Element of the carrier of V
the Source of V . b is Element of the carrier of V
Hom ((dom b),(cod (id C))) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom b & cod b₁ = cod (id C) ) } is set
[:(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C)))):] is Relation-like set
bool [:(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C)))):] is non empty set
[(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C))))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C))))} is non empty set
{(Hom ((cod b),(dom (id C))))} is non empty set
{{(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C))))},{(Hom ((cod b),(dom (id C))))}} is non empty set
(V,b,(id C)) is Relation-like Hom ((cod b),(dom (id C))) -defined Hom ((dom b),(cod (id C))) -valued Function-like quasi_total Element of bool [:(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C)))):]
[[(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C))))],(V,b,(id C))] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C)))):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C)))):]):] is Relation-like non empty set
{[(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C))))],(V,b,(id C))} is non empty set
{[(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C))))]} is Relation-like Function-like non empty set
{{[(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C))))],(V,b,(id C))},{[(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C))))]}} is non empty set
Hom ((cod b),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = cod b & cod b₁ = C ) } is set
Hom ((dom b),C) is Element of bool the carrier' of V
{ b₁ where b₁ is Element of the carrier' of V : ( dom b₁ = dom b & cod b₁ = C ) } is set
[:(Hom ((cod b),C)),(Hom ((dom b),C)):] is Relation-like set
bool [:(Hom ((cod b),C)),(Hom ((dom b),C)):] is non empty set
(V,C,b) is Relation-like Hom ((cod b),C) -defined Hom ((dom b),C) -valued Function-like quasi_total Element of bool [:(Hom ((cod b),C)),(Hom ((dom b),C)):]
[[(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C))))],(V,C,b)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod b),C)),(Hom ((dom b),C)):]):]
[:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod b),C)),(Hom ((dom b),C)):]):] is Relation-like non empty set
{[(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C))))],(V,C,b)} is non empty set
{{[(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C))))],(V,C,b)},{[(Hom ((cod b),(dom (id C)))),(Hom ((dom b),(cod (id C))))]}} is non empty set
[(Hom ((cod b),C)),(Hom ((dom b),(cod (id C))))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod b),C)),(Hom ((dom b),(cod (id C))))} is non empty set
{(Hom ((cod b),C))} is non empty set
{{(Hom ((cod b),C)),(Hom ((dom b),(cod (id C))))},{(Hom ((cod b),C))}} is non empty set
[[(Hom ((cod b),C)),(Hom ((dom b),(cod (id C))))],(V,C,b)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod b),C)),(Hom ((dom b),C)):]):]
{[(Hom ((cod b),C)),(Hom ((dom b),(cod (id C))))],(V,C,b)} is non empty set
{[(Hom ((cod b),C)),(Hom ((dom b),(cod (id C))))]} is Relation-like Function-like non empty set
{{[(Hom ((cod b),C)),(Hom ((dom b),(cod (id C))))],(V,C,b)},{[(Hom ((cod b),C)),(Hom ((dom b),(cod (id C))))]}} is non empty set
[(Hom ((cod b),C)),(Hom ((dom b),C))] is V15() Element of [:(bool the carrier' of V),(bool the carrier' of V):]
{(Hom ((cod b),C)),(Hom ((dom b),C))} is non empty set
{{(Hom ((cod b),C)),(Hom ((dom b),C))},{(Hom ((cod b),C))}} is non empty set
[[(Hom ((cod b),C)),(Hom ((dom b),C))],(V,C,b)] is V15() Element of [:[:(bool the carrier' of V),(bool the carrier' of V):],(bool [:(Hom ((cod b),C)),(Hom ((dom b),C)):]):]
{[(Hom ((cod b),C)),(Hom ((dom b),C))],(V,C,b)} is non empty set
{[(Hom ((cod b),C)),(Hom ((dom b),C))]} is Relation-like Function-like non empty set
{{[(Hom ((cod b),C)),(Hom ((dom b),C))],(V,C,b)},{[(Hom ((cod b),C)),(Hom ((dom b),C))]}} is non empty set
(V,C) . b is Element of ((V))
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of C is non empty set
(C) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
[:C,C:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of C, the carrier of C:] is Relation-like non empty set
the carrier' of C is non empty set
[: the carrier' of C, the carrier' of C:] is Relation-like non empty set
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
[: the Source of C, the Source of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is non empty set
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the Target of C, the Target of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
the Comp of C is Relation-like [: the carrier' of C, the carrier' of C:] -defined the carrier' of C -valued Function-like Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:]
[:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is non empty set
|: the Comp of C, the Comp of C:| is Relation-like [:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] -defined [: the carrier' of C, the carrier' of C:] -valued Function-like Element of bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
[:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is non empty set
CatStr(# [: the carrier of C, the carrier of C:],[: the carrier' of C, the carrier' of C:],[: the Source of C, the Source of C:],[: the Target of C, the Target of C:],|: the Comp of C, the Comp of C:| #) is non empty non void V59() strict CatStr
the carrier' of [:C,C:] is non empty set
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(C) is Relation-like the carrier' of [:C,C:] -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:C,C:],((C)):]
[: the carrier' of [:C,C:],((C)):] is Relation-like non empty set
bool [: the carrier' of [:C,C:],((C)):] is non empty set
a is Element of the carrier of C
id a is Morphism of a,a
b is Element of the carrier of C
Hom (a,b) is Element of bool the carrier' of C
bool the carrier' of C is non empty set
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = b ) } is set
id b is Morphism of b,b
[(id a),(id b)] is V15() Element of the carrier' of [:C,C:]
{(id a),(id b)} is non empty set
{(id a)} is non empty set
{{(id a),(id b)},{(id a)}} is non empty set
(C) . [(id a),(id b)] is Element of ((C))
(C,(id a),(id b)) is Relation-like Hom ((cod (id a)),(dom (id b))) -defined Hom ((dom (id a)),(cod (id b))) -valued Function-like quasi_total Element of bool [:(Hom ((cod (id a)),(dom (id b)))),(Hom ((dom (id a)),(cod (id b)))):]
cod (id a) is Element of the carrier of C
the Target of C . (id a) is Element of the carrier of C
dom (id b) is Element of the carrier of C
the Source of C . (id b) is Element of the carrier of C
Hom ((cod (id a)),(dom (id b))) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = cod (id a) & cod b₁ = dom (id b) ) } is set
dom (id a) is Element of the carrier of C
the Source of C . (id a) is Element of the carrier of C
cod (id b) is Element of the carrier of C
the Target of C . (id b) is Element of the carrier of C
Hom ((dom (id a)),(cod (id b))) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = dom (id a) & cod b₁ = cod (id b) ) } is set
[:(Hom ((cod (id a)),(dom (id b)))),(Hom ((dom (id a)),(cod (id b)))):] is Relation-like set
bool [:(Hom ((cod (id a)),(dom (id b)))),(Hom ((dom (id a)),(cod (id b)))):] is non empty set
A is Element of V
id A is Relation-like A -defined A -valued Function-like one-to-one total quasi_total Element of bool [:A,A:]
[:A,A:] is Relation-like set
bool [:A,A:] is non empty set
d is Element of the carrier of (V)
id d is Morphism of d,d
(V,d) is Element of V
(V,(V,d)) is Element of (V)
[:(V,d),(V,d):] is Relation-like set
bool [:(V,d),(V,d):] is non empty set
[(V,d),(V,d)] is V15() Element of [:V,V:]
{(V,d),(V,d)} is non empty set
{(V,d)} is non empty set
{{(V,d),(V,d)},{(V,d)}} is non empty set
id (V,d) is Relation-like (V,d) -defined (V,d) -valued Function-like one-to-one total quasi_total Element of bool [:(V,d),(V,d):]
[[(V,d),(V,d)],(id (V,d))] is V15() Element of [:[:V,V:],(bool [:(V,d),(V,d):]):]
[:[:V,V:],(bool [:(V,d),(V,d):]):] is Relation-like non empty set
{[(V,d),(V,d)],(id (V,d))} is non empty set
{[(V,d),(V,d)]} is Relation-like Function-like non empty set
{{[(V,d),(V,d)],(id (V,d))},{[(V,d),(V,d)]}} is non empty set
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
(C) is non empty set
the carrier of C is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
(C) is Relation-like the carrier' of [:C,C:] -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:C,C:],((C)):]
[:C,C:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of C, the carrier of C:] is Relation-like non empty set
the carrier' of C is non empty set
[: the carrier' of C, the carrier' of C:] is Relation-like non empty set
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
[: the Source of C, the Source of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is non empty set
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the Target of C, the Target of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
the Comp of C is Relation-like [: the carrier' of C, the carrier' of C:] -defined the carrier' of C -valued Function-like Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:]
[:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is non empty set
|: the Comp of C, the Comp of C:| is Relation-like [:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] -defined [: the carrier' of C, the carrier' of C:] -valued Function-like Element of bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
[:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is non empty set
CatStr(# [: the carrier of C, the carrier of C:],[: the carrier' of C, the carrier' of C:],[: the Source of C, the Source of C:],[: the Target of C, the Target of C:],|: the Comp of C, the Comp of C:| #) is non empty non void V59() strict CatStr
the carrier' of [:C,C:] is non empty set
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
[: the carrier' of [:C,C:],((C)):] is Relation-like non empty set
bool [: the carrier' of [:C,C:],((C)):] is non empty set
C opp is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
~ the Comp of C is Relation-like [: the carrier' of C, the carrier' of C:] -defined the carrier' of C -valued Function-like Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:]
CatStr(# the carrier of C, the carrier' of C, the Target of C, the Source of C,(~ the Comp of C) #) is non empty non void V59() strict CatStr
[:(C opp),C:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of (C opp) is non empty set
[: the carrier of (C opp), the carrier of C:] is Relation-like non empty set
the carrier' of (C opp) is non empty set
[: the carrier' of (C opp), the carrier' of C:] is Relation-like non empty set
the Source of (C opp) is Relation-like the carrier' of (C opp) -defined the carrier of (C opp) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (C opp), the carrier of (C opp):]
[: the carrier' of (C opp), the carrier of (C opp):] is Relation-like non empty set
bool [: the carrier' of (C opp), the carrier of (C opp):] is non empty set
[: the Source of (C opp), the Source of C:] is Relation-like [: the carrier' of (C opp), the carrier' of C:] -defined [: the carrier of (C opp), the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:]
[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:] is Relation-like non empty set
bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:] is non empty set
the Target of (C opp) is Relation-like the carrier' of (C opp) -defined the carrier of (C opp) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (C opp), the carrier of (C opp):]
[: the Target of (C opp), the Target of C:] is Relation-like [: the carrier' of (C opp), the carrier' of C:] -defined [: the carrier of (C opp), the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:]
the Comp of (C opp) is Relation-like [: the carrier' of (C opp), the carrier' of (C opp):] -defined the carrier' of (C opp) -valued Function-like Element of bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):]
[: the carrier' of (C opp), the carrier' of (C opp):] is Relation-like non empty set
[:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):] is Relation-like non empty set
bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):] is non empty set
|: the Comp of (C opp), the Comp of C:| is Relation-like [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:] -defined [: the carrier' of (C opp), the carrier' of C:] -valued Function-like Element of bool [:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:]
[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:] is Relation-like non empty set
[:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:] is Relation-like non empty set
bool [:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:] is non empty set
CatStr(# [: the carrier of (C opp), the carrier of C:],[: the carrier' of (C opp), the carrier' of C:],[: the Source of (C opp), the Source of C:],[: the Target of (C opp), the Target of C:],|: the Comp of (C opp), the Comp of C:| #) is non empty non void V59() strict CatStr
the carrier' of [:(C opp),C:] is non empty set
[: the carrier' of [:(C opp),C:], the carrier' of (V):] is Relation-like non empty set
bool [: the carrier' of [:(C opp),C:], the carrier' of (V):] is non empty set
the carrier of [:(C opp),C:] is non empty set
the carrier of (V) is non empty set
a is Relation-like the carrier' of [:(C opp),C:] -defined the carrier' of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:(C opp),C:], the carrier' of (V):]
b is Element of the carrier of [:(C opp),C:]
id b is Morphism of b,b
a . (id b) is Element of the carrier' of (V)
A is Element of the carrier of (C opp)
d is Element of the carrier of C
[A,d] is V15() Element of the carrier of [:(C opp),C:]
{A,d} is non empty set
{A} is non empty set
{{A,d},{A}} is non empty set
opp A is Element of the carrier of C
A opp is Element of the carrier of ((C opp) opp)
(C opp) opp is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
~ the Comp of (C opp) is Relation-like [: the carrier' of (C opp), the carrier' of (C opp):] -defined the carrier' of (C opp) -valued Function-like Element of bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):]
CatStr(# the carrier of (C opp), the carrier' of (C opp), the Target of (C opp), the Source of (C opp),(~ the Comp of (C opp)) #) is non empty non void V59() strict CatStr
the carrier of ((C opp) opp) is non empty set
Hom ((opp A),d) is Element of bool the carrier' of C
bool the carrier' of C is non empty set
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = opp A & cod b₁ = d ) } is set
g9 is Element of V
(V,g9) is Element of the carrier of (V)
f is Element of the carrier of (V)
id f is Morphism of f,f
id (opp A) is Morphism of opp A, opp A
id A is Morphism of A,A
id d is Morphism of d,d
[(id (opp A)),(id d)] is V15() Element of the carrier' of [:C,C:]
{(id (opp A)),(id d)} is non empty set
{(id (opp A))} is non empty set
{{(id (opp A)),(id d)},{(id (opp A))}} is non empty set
b is Element of the carrier' of [:(C opp),C:]
dom b is Element of the carrier of [:(C opp),C:]
the Source of [:(C opp),C:] is Relation-like the carrier' of [:(C opp),C:] -defined the carrier of [:(C opp),C:] -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:(C opp),C:], the carrier of [:(C opp),C:]:]
[: the carrier' of [:(C opp),C:], the carrier of [:(C opp),C:]:] is Relation-like non empty set
bool [: the carrier' of [:(C opp),C:], the carrier of [:(C opp),C:]:] is non empty set
the Source of [:(C opp),C:] . b is Element of the carrier of [:(C opp),C:]
id (dom b) is Morphism of dom b, dom b
a . (id (dom b)) is Element of the carrier' of (V)
a . b is Element of the carrier' of (V)
dom (a . b) is Element of the carrier of (V)
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Source of (V) . (a . b) is Element of the carrier of (V)
id (dom (a . b)) is Morphism of dom (a . b), dom (a . b)
cod b is Element of the carrier of [:(C opp),C:]
the Target of [:(C opp),C:] is Relation-like the carrier' of [:(C opp),C:] -defined the carrier of [:(C opp),C:] -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:(C opp),C:], the carrier of [:(C opp),C:]:]
the Target of [:(C opp),C:] . b is Element of the carrier of [:(C opp),C:]
id (cod b) is Morphism of cod b, cod b
a . (id (cod b)) is Element of the carrier' of (V)
cod (a . b) is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Target of (V) . (a . b) is Element of the carrier of (V)
id (cod (a . b)) is Morphism of cod (a . b), cod (a . b)
A is Element of the carrier' of (C opp)
d is Element of the carrier' of C
[A,d] is V15() Element of the carrier' of [:(C opp),C:]
{A,d} is non empty set
{A} is non empty set
{{A,d},{A}} is non empty set
opp A is Element of the carrier' of C
A opp is Element of the carrier' of ((C opp) opp)
(C opp) opp is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
~ the Comp of (C opp) is Relation-like [: the carrier' of (C opp), the carrier' of (C opp):] -defined the carrier' of (C opp) -valued Function-like Element of bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):]
CatStr(# the carrier of (C opp), the carrier' of (C opp), the Target of (C opp), the Source of (C opp),(~ the Comp of (C opp)) #) is non empty non void V59() strict CatStr
the carrier' of ((C opp) opp) is non empty set
cod (opp A) is Element of the carrier of C
the Target of C . (opp A) is Element of the carrier of C
dom d is Element of the carrier of C
the Source of C . d is Element of the carrier of C
Hom ((cod (opp A)),(dom d)) is Element of bool the carrier' of C
bool the carrier' of C is non empty set
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = cod (opp A) & cod b₁ = dom d ) } is set
dom (opp A) is Element of the carrier of C
the Source of C . (opp A) is Element of the carrier of C
cod d is Element of the carrier of C
the Target of C . d is Element of the carrier of C
Hom ((dom (opp A)),(cod d)) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = dom (opp A) & cod b₁ = cod d ) } is set
dom A is Element of the carrier of (C opp)
the Source of (C opp) . A is Element of the carrier of (C opp)
opp (dom A) is Element of the carrier of C
(dom A) opp is Element of the carrier of ((C opp) opp)
the carrier of ((C opp) opp) is non empty set
id (opp (dom A)) is Morphism of opp (dom A), opp (dom A)
id (dom A) is Morphism of dom A, dom A
cod A is Element of the carrier of (C opp)
the Target of (C opp) . A is Element of the carrier of (C opp)
opp (cod A) is Element of the carrier of C
(cod A) opp is Element of the carrier of ((C opp) opp)
id (opp (cod A)) is Morphism of opp (cod A), opp (cod A)
id (cod A) is Morphism of cod A, cod A
[:(bool the carrier' of C),(bool the carrier' of C):] is Relation-like non empty set
[:(Hom ((cod (opp A)),(dom d))),(Hom ((dom (opp A)),(cod d))):] is Relation-like set
bool [:(Hom ((cod (opp A)),(dom d))),(Hom ((dom (opp A)),(cod d))):] is non empty set
[(Hom ((cod (opp A)),(dom d))),(Hom ((dom (opp A)),(cod d)))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom ((cod (opp A)),(dom d))),(Hom ((dom (opp A)),(cod d)))} is non empty set
{(Hom ((cod (opp A)),(dom d)))} is non empty set
{{(Hom ((cod (opp A)),(dom d))),(Hom ((dom (opp A)),(cod d)))},{(Hom ((cod (opp A)),(dom d)))}} is non empty set
(C,(opp A),d) is Relation-like Hom ((cod (opp A)),(dom d)) -defined Hom ((dom (opp A)),(cod d)) -valued Function-like quasi_total Element of bool [:(Hom ((cod (opp A)),(dom d))),(Hom ((dom (opp A)),(cod d))):]
[[(Hom ((cod (opp A)),(dom d))),(Hom ((dom (opp A)),(cod d)))],(C,(opp A),d)] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod (opp A)),(dom d))),(Hom ((dom (opp A)),(cod d))):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod (opp A)),(dom d))),(Hom ((dom (opp A)),(cod d))):]):] is Relation-like non empty set
{[(Hom ((cod (opp A)),(dom d))),(Hom ((dom (opp A)),(cod d)))],(C,(opp A),d)} is non empty set
{[(Hom ((cod (opp A)),(dom d))),(Hom ((dom (opp A)),(cod d)))]} is Relation-like Function-like non empty set
{{[(Hom ((cod (opp A)),(dom d))),(Hom ((dom (opp A)),(cod d)))],(C,(opp A),d)},{[(Hom ((cod (opp A)),(dom d))),(Hom ((dom (opp A)),(cod d)))]}} is non empty set
(V,(a . b)) is Element of (V)
(V,(V,(a . b))) is Element of V
(V,(a . b)) `1 is set
((V,(a . b)) `1) `1 is set
(V,(V,(a . b))) is Element of V
((V,(a . b)) `1) `2 is set
[(V,(V,(a . b))),(V,(V,(a . b)))] is V15() Element of [:V,V:]
{(V,(V,(a . b))),(V,(V,(a . b)))} is non empty set
{(V,(V,(a . b)))} is non empty set
{{(V,(V,(a . b))),(V,(V,(a . b)))},{(V,(V,(a . b)))}} is non empty set
(V,(a . b)) `2 is Relation-like Function-like set
[[(V,(V,(a . b))),(V,(V,(a . b)))],((V,(a . b)) `2)] is V15() set
{[(V,(V,(a . b))),(V,(V,(a . b)))],((V,(a . b)) `2)} is non empty set
{[(V,(V,(a . b))),(V,(V,(a . b)))]} is Relation-like Function-like non empty set
{{[(V,(V,(a . b))),(V,(V,(a . b)))],((V,(a . b)) `2)},{[(V,(V,(a . b))),(V,(V,(a . b)))]}} is non empty set
[(dom (a . b)),(V,(V,(a . b)))] is V15() Element of [: the carrier of (V),V:]
[: the carrier of (V),V:] is Relation-like non empty set
{(dom (a . b)),(V,(V,(a . b)))} is non empty set
{(dom (a . b))} is non empty set
{{(dom (a . b)),(V,(V,(a . b)))},{(dom (a . b))}} is non empty set
[[(dom (a . b)),(V,(V,(a . b)))],((V,(a . b)) `2)] is V15() set
{[(dom (a . b)),(V,(V,(a . b)))],((V,(a . b)) `2)} is non empty set
{[(dom (a . b)),(V,(V,(a . b)))]} is Relation-like Function-like non empty set
{{[(dom (a . b)),(V,(V,(a . b)))],((V,(a . b)) `2)},{[(dom (a . b)),(V,(V,(a . b)))]}} is non empty set
[(dom (a . b)),(cod (a . b))] is V15() Element of the carrier of [:(V),(V):]
[:(V),(V):] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of (V), the carrier of (V):] is Relation-like non empty set
[: the carrier' of (V), the carrier' of (V):] is Relation-like non empty set
[: the Source of (V), the Source of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is non empty set
[: the Target of (V), the Target of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
the Comp of (V) is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined the carrier' of (V) -valued Function-like Element of bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):]
[:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is non empty set
|: the Comp of (V), the Comp of (V):| is Relation-like [:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] -defined [: the carrier' of (V), the carrier' of (V):] -valued Function-like Element of bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
[:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is non empty set
CatStr(# [: the carrier of (V), the carrier of (V):],[: the carrier' of (V), the carrier' of (V):],[: the Source of (V), the Source of (V):],[: the Target of (V), the Target of (V):],|: the Comp of (V), the Comp of (V):| #) is non empty non void V59() strict CatStr
the carrier of [:(V),(V):] is non empty set
{(dom (a . b)),(cod (a . b))} is non empty set
{{(dom (a . b)),(cod (a . b))},{(dom (a . b))}} is non empty set
[[(dom (a . b)),(cod (a . b))],((V,(a . b)) `2)] is V15() set
{[(dom (a . b)),(cod (a . b))],((V,(a . b)) `2)} is non empty set
{[(dom (a . b)),(cod (a . b))]} is Relation-like Function-like non empty set
{{[(dom (a . b)),(cod (a . b))],((V,(a . b)) `2)},{[(dom (a . b)),(cod (a . b))]}} is non empty set
[(dom A),(dom d)] is V15() Element of the carrier of [:(C opp),C:]
{(dom A),(dom d)} is non empty set
{(dom A)} is non empty set
{{(dom A),(dom d)},{(dom A)}} is non empty set
id [(dom A),(dom d)] is Morphism of [(dom A),(dom d)],[(dom A),(dom d)]
a . (id [(dom A),(dom d)]) is Element of the carrier' of (V)
id (dom d) is Morphism of dom d, dom d
[(id (dom A)),(id (dom d))] is V15() Element of the carrier' of [:(C opp),C:]
{(id (dom A)),(id (dom d))} is non empty set
{(id (dom A))} is non empty set
{{(id (dom A)),(id (dom d))},{(id (dom A))}} is non empty set
a . [(id (dom A)),(id (dom d))] is Element of the carrier' of (V)
g9 is Element of V
(V,g9) is Element of the carrier of (V)
id (V,g9) is Morphism of (V,g9),(V,g9)
[(cod A),(cod d)] is V15() Element of the carrier of [:(C opp),C:]
{(cod A),(cod d)} is non empty set
{(cod A)} is non empty set
{{(cod A),(cod d)},{(cod A)}} is non empty set
id [(cod A),(cod d)] is Morphism of [(cod A),(cod d)],[(cod A),(cod d)]
a . (id [(cod A),(cod d)]) is Element of the carrier' of (V)
id (cod d) is Morphism of cod d, cod d
[(id (cod A)),(id (cod d))] is V15() Element of the carrier' of [:(C opp),C:]
{(id (cod A)),(id (cod d))} is non empty set
{(id (cod A))} is non empty set
{{(id (cod A)),(id (cod d))},{(id (cod A))}} is non empty set
a . [(id (cod A)),(id (cod d))] is Element of the carrier' of (V)
f is Element of V
(V,f) is Element of the carrier of (V)
id (V,f) is Morphism of (V,f),(V,f)
A is Element of the carrier' of [:(C opp),C:]
dom A is Element of the carrier of [:(C opp),C:]
the Source of [:(C opp),C:] is Relation-like the carrier' of [:(C opp),C:] -defined the carrier of [:(C opp),C:] -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:(C opp),C:], the carrier of [:(C opp),C:]:]
[: the carrier' of [:(C opp),C:], the carrier of [:(C opp),C:]:] is Relation-like non empty set
bool [: the carrier' of [:(C opp),C:], the carrier of [:(C opp),C:]:] is non empty set
the Source of [:(C opp),C:] . A is Element of the carrier of [:(C opp),C:]
b is Element of the carrier' of [:(C opp),C:]
cod b is Element of the carrier of [:(C opp),C:]
the Target of [:(C opp),C:] is Relation-like the carrier' of [:(C opp),C:] -defined the carrier of [:(C opp),C:] -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:(C opp),C:], the carrier of [:(C opp),C:]:]
the Target of [:(C opp),C:] . b is Element of the carrier of [:(C opp),C:]
d is Element of the carrier' of (C opp)
g9 is Element of the carrier' of C
[d,g9] is V15() Element of the carrier' of [:(C opp),C:]
{d,g9} is non empty set
{d} is non empty set
{{d,g9},{d}} is non empty set
bool the carrier' of C is non empty set
[:(bool the carrier' of C),(bool the carrier' of C):] is Relation-like non empty set
opp d is Element of the carrier' of C
d opp is Element of the carrier' of ((C opp) opp)
(C opp) opp is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
~ the Comp of (C opp) is Relation-like [: the carrier' of (C opp), the carrier' of (C opp):] -defined the carrier' of (C opp) -valued Function-like Element of bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):]
CatStr(# the carrier of (C opp), the carrier' of (C opp), the Target of (C opp), the Source of (C opp),(~ the Comp of (C opp)) #) is non empty non void V59() strict CatStr
the carrier' of ((C opp) opp) is non empty set
cod (opp d) is Element of the carrier of C
the Target of C . (opp d) is Element of the carrier of C
dom g9 is Element of the carrier of C
the Source of C . g9 is Element of the carrier of C
Hom ((cod (opp d)),(dom g9)) is Element of bool the carrier' of C
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of C : ( dom b<sub>1</sub> = cod (opp d) & cod b<sub>1</sub> = dom g9 ) } is set
dom (opp d) is Element of the carrier of C
the Source of C . (opp d) is Element of the carrier of C
cod g9 is Element of the carrier of C
the Target of C . g9 is Element of the carrier of C
Hom ((dom (opp d)),(cod g9)) is Element of bool the carrier' of C
{ b<sub>1</sub> where b<sub>1</sub> is Element of the carrier' of C : ( dom b<sub>1</sub> = dom (opp d) & cod b<sub>1</sub> = cod g9 ) } is set
[:(Hom ((cod (opp d)),(dom g9))),(Hom ((dom (opp d)),(cod g9))):] is Relation-like set
bool [:(Hom ((cod (opp d)),(dom g9))),(Hom ((dom (opp d)),(cod g9))):] is non empty set
[(Hom ((cod (opp d)),(dom g9))),(Hom ((dom (opp d)),(cod g9)))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom ((cod (opp d)),(dom g9))),(Hom ((dom (opp d)),(cod g9)))} is non empty set
{(Hom ((cod (opp d)),(dom g9)))} is non empty set
{{(Hom ((cod (opp d)),(dom g9))),(Hom ((dom (opp d)),(cod g9)))},{(Hom ((cod (opp d)),(dom g9)))}} is non empty set
(C,(opp d),g9) is Relation-like Hom ((cod (opp d)),(dom g9)) -defined Hom ((dom (opp d)),(cod g9)) -valued Function-like quasi_total Element of bool [:(Hom ((cod (opp d)),(dom g9))),(Hom ((dom (opp d)),(cod g9))):]
[[(Hom ((cod (opp d)),(dom g9))),(Hom ((dom (opp d)),(cod g9)))],(C,(opp d),g9)] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod (opp d)),(dom g9))),(Hom ((dom (opp d)),(cod g9))):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod (opp d)),(dom g9))),(Hom ((dom (opp d)),(cod g9))):]):] is Relation-like non empty set
{[(Hom ((cod (opp d)),(dom g9))),(Hom ((dom (opp d)),(cod g9)))],(C,(opp d),g9)} is non empty set
{[(Hom ((cod (opp d)),(dom g9))),(Hom ((dom (opp d)),(cod g9)))]} is Relation-like Function-like non empty set
{{[(Hom ((cod (opp d)),(dom g9))),(Hom ((dom (opp d)),(cod g9)))],(C,(opp d),g9)},{[(Hom ((cod (opp d)),(dom g9))),(Hom ((dom (opp d)),(cod g9)))]}} is non empty set
a . A is Element of the carrier' of (V)
(V,(a . A)) is Element of (V)
(V,(V,(a . A))) is Element of V
(V,(a . A)) `1 is set
((V,(a . A)) `1) `1 is set
(V,(V,(a . A))) is Element of V
((V,(a . A)) `1) `2 is set
[(V,(V,(a . A))),(V,(V,(a . A)))] is V15() Element of [:V,V:]
{(V,(V,(a . A))),(V,(V,(a . A)))} is non empty set
{(V,(V,(a . A)))} is non empty set
{{(V,(V,(a . A))),(V,(V,(a . A)))},{(V,(V,(a . A)))}} is non empty set
(V,(a . A)) `2 is Relation-like Function-like set
[[(V,(V,(a . A))),(V,(V,(a . A)))],((V,(a . A)) `2)] is V15() set
{[(V,(V,(a . A))),(V,(V,(a . A)))],((V,(a . A)) `2)} is non empty set
{[(V,(V,(a . A))),(V,(V,(a . A)))]} is Relation-like Function-like non empty set
{{[(V,(V,(a . A))),(V,(V,(a . A)))],((V,(a . A)) `2)},{[(V,(V,(a . A))),(V,(V,(a . A)))]}} is non empty set
dom (a . A) is Element of the carrier of (V)
the Source of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
[: the carrier' of (V), the carrier of (V):] is Relation-like non empty set
bool [: the carrier' of (V), the carrier of (V):] is non empty set
the Source of (V) . (a . A) is Element of the carrier of (V)
[(dom (a . A)),(V,(V,(a . A)))] is V15() Element of [: the carrier of (V),V:]
[: the carrier of (V),V:] is Relation-like non empty set
{(dom (a . A)),(V,(V,(a . A)))} is non empty set
{(dom (a . A))} is non empty set
{{(dom (a . A)),(V,(V,(a . A)))},{(dom (a . A))}} is non empty set
[[(dom (a . A)),(V,(V,(a . A)))],((V,(a . A)) `2)] is V15() set
{[(dom (a . A)),(V,(V,(a . A)))],((V,(a . A)) `2)} is non empty set
{[(dom (a . A)),(V,(V,(a . A)))]} is Relation-like Function-like non empty set
{{[(dom (a . A)),(V,(V,(a . A)))],((V,(a . A)) `2)},{[(dom (a . A)),(V,(V,(a . A)))]}} is non empty set
cod (a . A) is Element of the carrier of (V)
the Target of (V) is Relation-like the carrier' of (V) -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (V), the carrier of (V):]
the Target of (V) . (a . A) is Element of the carrier of (V)
[(dom (a . A)),(cod (a . A))] is V15() Element of the carrier of [:(V),(V):]
[:(V),(V):] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of (V), the carrier of (V):] is Relation-like non empty set
[: the carrier' of (V), the carrier' of (V):] is Relation-like non empty set
[: the Source of (V), the Source of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:] is non empty set
[: the Target of (V), the Target of (V):] is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined [: the carrier of (V), the carrier of (V):] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (V), the carrier' of (V):],[: the carrier of (V), the carrier of (V):]:]
the Comp of (V) is Relation-like [: the carrier' of (V), the carrier' of (V):] -defined the carrier' of (V) -valued Function-like Element of bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):]
[:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is Relation-like non empty set
bool [:[: the carrier' of (V), the carrier' of (V):], the carrier' of (V):] is non empty set
|: the Comp of (V), the Comp of (V):| is Relation-like [:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] -defined [: the carrier' of (V), the carrier' of (V):] -valued Function-like Element of bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:]
[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
[:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is Relation-like non empty set
bool [:[:[: the carrier' of (V), the carrier' of (V):],[: the carrier' of (V), the carrier' of (V):]:],[: the carrier' of (V), the carrier' of (V):]:] is non empty set
CatStr(# [: the carrier of (V), the carrier of (V):],[: the carrier' of (V), the carrier' of (V):],[: the Source of (V), the Source of (V):],[: the Target of (V), the Target of (V):],|: the Comp of (V), the Comp of (V):| #) is non empty non void V59() strict CatStr
the carrier of [:(V),(V):] is non empty set
{(dom (a . A)),(cod (a . A))} is non empty set
{{(dom (a . A)),(cod (a . A))},{(dom (a . A))}} is non empty set
[[(dom (a . A)),(cod (a . A))],((V,(a . A)) `2)] is V15() set
{[(dom (a . A)),(cod (a . A))],((V,(a . A)) `2)} is non empty set
{[(dom (a . A)),(cod (a . A))]} is Relation-like Function-like non empty set
{{[(dom (a . A)),(cod (a . A))],((V,(a . A)) `2)},{[(dom (a . A)),(cod (a . A))]}} is non empty set
f is Element of the carrier' of (C opp)
f9 is Element of the carrier' of C
[f,f9] is V15() Element of the carrier' of [:(C opp),C:]
{f,f9} is non empty set
{f} is non empty set
{{f,f9},{f}} is non empty set
opp f is Element of the carrier' of C
f opp is Element of the carrier' of ((C opp) opp)
cod (opp f) is Element of the carrier of C
the Target of C . (opp f) is Element of the carrier of C
dom f9 is Element of the carrier of C
the Source of C . f9 is Element of the carrier of C
Hom ((cod (opp f)),(dom f9)) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = cod (opp f) & cod b₁ = dom f9 ) } is set
dom (opp f) is Element of the carrier of C
the Source of C . (opp f) is Element of the carrier of C
cod f9 is Element of the carrier of C
the Target of C . f9 is Element of the carrier of C
Hom ((dom (opp f)),(cod f9)) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = dom (opp f) & cod b₁ = cod f9 ) } is set
[:(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp f)),(cod f9))):] is Relation-like set
bool [:(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp f)),(cod f9))):] is non empty set
[(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp f)),(cod f9)))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp f)),(cod f9)))} is non empty set
{(Hom ((cod (opp f)),(dom f9)))} is non empty set
{{(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp f)),(cod f9)))},{(Hom ((cod (opp f)),(dom f9)))}} is non empty set
(C,(opp f),f9) is Relation-like Hom ((cod (opp f)),(dom f9)) -defined Hom ((dom (opp f)),(cod f9)) -valued Function-like quasi_total Element of bool [:(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp f)),(cod f9))):]
[[(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp f)),(cod f9)))],(C,(opp f),f9)] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp f)),(cod f9))):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp f)),(cod f9))):]):] is Relation-like non empty set
{[(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp f)),(cod f9)))],(C,(opp f),f9)} is non empty set
{[(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp f)),(cod f9)))]} is Relation-like Function-like non empty set
{{[(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp f)),(cod f9)))],(C,(opp f),f9)},{[(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp f)),(cod f9)))]}} is non empty set
a . b is Element of the carrier' of (V)
(V,(a . b)) is Element of (V)
(V,(V,(a . b))) is Element of V
(V,(a . b)) `1 is set
((V,(a . b)) `1) `1 is set
(V,(V,(a . b))) is Element of V
((V,(a . b)) `1) `2 is set
[(V,(V,(a . b))),(V,(V,(a . b)))] is V15() Element of [:V,V:]
{(V,(V,(a . b))),(V,(V,(a . b)))} is non empty set
{(V,(V,(a . b)))} is non empty set
{{(V,(V,(a . b))),(V,(V,(a . b)))},{(V,(V,(a . b)))}} is non empty set
(V,(a . b)) `2 is Relation-like Function-like set
[[(V,(V,(a . b))),(V,(V,(a . b)))],((V,(a . b)) `2)] is V15() set
{[(V,(V,(a . b))),(V,(V,(a . b)))],((V,(a . b)) `2)} is non empty set
{[(V,(V,(a . b))),(V,(V,(a . b)))]} is Relation-like Function-like non empty set
{{[(V,(V,(a . b))),(V,(V,(a . b)))],((V,(a . b)) `2)},{[(V,(V,(a . b))),(V,(V,(a . b)))]}} is non empty set
dom (a . b) is Element of the carrier of (V)
the Source of (V) . (a . b) is Element of the carrier of (V)
[(dom (a . b)),(V,(V,(a . b)))] is V15() Element of [: the carrier of (V),V:]
{(dom (a . b)),(V,(V,(a . b)))} is non empty set
{(dom (a . b))} is non empty set
{{(dom (a . b)),(V,(V,(a . b)))},{(dom (a . b))}} is non empty set
[[(dom (a . b)),(V,(V,(a . b)))],((V,(a . b)) `2)] is V15() set
{[(dom (a . b)),(V,(V,(a . b)))],((V,(a . b)) `2)} is non empty set
{[(dom (a . b)),(V,(V,(a . b)))]} is Relation-like Function-like non empty set
{{[(dom (a . b)),(V,(V,(a . b)))],((V,(a . b)) `2)},{[(dom (a . b)),(V,(V,(a . b)))]}} is non empty set
cod (a . b) is Element of the carrier of (V)
the Target of (V) . (a . b) is Element of the carrier of (V)
[(dom (a . b)),(cod (a . b))] is V15() Element of the carrier of [:(V),(V):]
{(dom (a . b)),(cod (a . b))} is non empty set
{{(dom (a . b)),(cod (a . b))},{(dom (a . b))}} is non empty set
[[(dom (a . b)),(cod (a . b))],((V,(a . b)) `2)] is V15() set
{[(dom (a . b)),(cod (a . b))],((V,(a . b)) `2)} is non empty set
{[(dom (a . b)),(cod (a . b))]} is Relation-like Function-like non empty set
{{[(dom (a . b)),(cod (a . b))],((V,(a . b)) `2)},{[(dom (a . b)),(cod (a . b))]}} is non empty set
cod f is Element of the carrier of (C opp)
the Target of (C opp) . f is Element of the carrier of (C opp)
[(cod f),(cod f9)] is V15() Element of the carrier of [:(C opp),C:]
{(cod f),(cod f9)} is non empty set
{(cod f)} is non empty set
{{(cod f),(cod f9)},{(cod f)}} is non empty set
dom d is Element of the carrier of (C opp)
the Source of (C opp) . d is Element of the carrier of (C opp)
[(dom d),(dom g9)] is V15() Element of the carrier of [:(C opp),C:]
{(dom d),(dom g9)} is non empty set
{(dom d)} is non empty set
{{(dom d),(dom g9)},{(dom d)}} is non empty set
(opp f) (*) (opp d) is Element of the carrier' of C
dom ((opp f) (*) (opp d)) is Element of the carrier of C
the Source of C . ((opp f) (*) (opp d)) is Element of the carrier of C
cod ((opp f) (*) (opp d)) is Element of the carrier of C
the Target of C . ((opp f) (*) (opp d)) is Element of the carrier of C
g9 (*) f9 is Element of the carrier' of C
dom (g9 (*) f9) is Element of the carrier of C
the Source of C . (g9 (*) f9) is Element of the carrier of C
cod (g9 (*) f9) is Element of the carrier of C
the Target of C . (g9 (*) f9) is Element of the carrier of C
A (*) b is Element of the carrier' of [:(C opp),C:]
a . (A (*) b) is Element of the carrier' of (V)
d (*) f is Element of the carrier' of (C opp)
opp (d (*) f) is Element of the carrier' of C
(d (*) f) opp is Element of the carrier' of ((C opp) opp)
[(opp (d (*) f)),(g9 (*) f9)] is V15() Element of the carrier' of [:C,C:]
{(opp (d (*) f)),(g9 (*) f9)} is non empty set
{(opp (d (*) f))} is non empty set
{{(opp (d (*) f)),(g9 (*) f9)},{(opp (d (*) f))}} is non empty set
a . [(opp (d (*) f)),(g9 (*) f9)] is set
[((opp f) (*) (opp d)),(g9 (*) f9)] is V15() Element of the carrier' of [:C,C:]
{((opp f) (*) (opp d)),(g9 (*) f9)} is non empty set
{((opp f) (*) (opp d))} is non empty set
{{((opp f) (*) (opp d)),(g9 (*) f9)},{((opp f) (*) (opp d))}} is non empty set
a . [((opp f) (*) (opp d)),(g9 (*) f9)] is set
Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9))) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = cod ((opp f) (*) (opp d)) & cod b₁ = dom (g9 (*) f9) ) } is set
Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9))) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = dom ((opp f) (*) (opp d)) & cod b₁ = cod (g9 (*) f9) ) } is set
[:(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9)))),(Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9)))):] is Relation-like set
bool [:(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9)))),(Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9)))):] is non empty set
[(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9)))),(Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9))))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9)))),(Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9))))} is non empty set
{(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9))))} is non empty set
{{(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9)))),(Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9))))},{(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9))))}} is non empty set
(C,((opp f) (*) (opp d)),(g9 (*) f9)) is Relation-like Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9))) -defined Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9))) -valued Function-like quasi_total Element of bool [:(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9)))),(Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9)))):]
[[(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9)))),(Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9))))],(C,((opp f) (*) (opp d)),(g9 (*) f9))] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9)))),(Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9)))):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9)))),(Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9)))):]):] is Relation-like non empty set
{[(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9)))),(Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9))))],(C,((opp f) (*) (opp d)),(g9 (*) f9))} is non empty set
{[(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9)))),(Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9))))]} is Relation-like Function-like non empty set
{{[(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9)))),(Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9))))],(C,((opp f) (*) (opp d)),(g9 (*) f9))},{[(Hom ((cod ((opp f) (*) (opp d))),(dom (g9 (*) f9)))),(Hom ((dom ((opp f) (*) (opp d))),(cod (g9 (*) f9))))]}} is non empty set
[:(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp d)),(cod g9))):] is Relation-like set
bool [:(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp d)),(cod g9))):] is non empty set
[(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp d)),(cod g9)))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp d)),(cod g9)))} is non empty set
{{(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp d)),(cod g9)))},{(Hom ((cod (opp f)),(dom f9)))}} is non empty set
(C,(opp d),g9) * (C,(opp f),f9) is Relation-like Hom ((cod (opp f)),(dom f9)) -defined Hom ((dom (opp d)),(cod g9)) -valued Function-like Element of bool [:(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp d)),(cod g9))):]
[[(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp d)),(cod g9)))],((C,(opp d),g9) * (C,(opp f),f9))] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp d)),(cod g9))):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp d)),(cod g9))):]):] is Relation-like non empty set
{[(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp d)),(cod g9)))],((C,(opp d),g9) * (C,(opp f),f9))} is non empty set
{[(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp d)),(cod g9)))]} is Relation-like Function-like non empty set
{{[(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp d)),(cod g9)))],((C,(opp d),g9) * (C,(opp f),f9))},{[(Hom ((cod (opp f)),(dom f9))),(Hom ((dom (opp d)),(cod g9)))]}} is non empty set
(V,(V,(a . b)),(V,(a . A))) is Element of (V)
(a . A) (*) (a . b) is Element of the carrier' of (V)
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of C is non empty set
(C) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
V is non empty set
a is Element of the carrier of C
(C,a) is Relation-like the carrier' of C -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C,((C)):]
the carrier' of C is non empty set
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
[: the carrier' of C,((C)):] is Relation-like non empty set
bool [: the carrier' of C,((C)):] is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
(C,a) is Relation-like the carrier' of C -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C,((C)):]
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
(C) is non empty set
the carrier of C is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
V is non empty set
(C) is Relation-like the carrier' of [:C,C:] -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:C,C:],((C)):]
[:C,C:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of C, the carrier of C:] is Relation-like non empty set
the carrier' of C is non empty set
[: the carrier' of C, the carrier' of C:] is Relation-like non empty set
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
[: the Source of C, the Source of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is non empty set
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the Target of C, the Target of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
the Comp of C is Relation-like [: the carrier' of C, the carrier' of C:] -defined the carrier' of C -valued Function-like Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:]
[:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is non empty set
|: the Comp of C, the Comp of C:| is Relation-like [:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] -defined [: the carrier' of C, the carrier' of C:] -valued Function-like Element of bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
[:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is non empty set
CatStr(# [: the carrier of C, the carrier of C:],[: the carrier' of C, the carrier' of C:],[: the Source of C, the Source of C:],[: the Target of C, the Target of C:],|: the Comp of C, the Comp of C:| #) is non empty non void V59() strict CatStr
the carrier' of [:C,C:] is non empty set
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
[: the carrier' of [:C,C:],((C)):] is Relation-like non empty set
bool [: the carrier' of [:C,C:],((C)):] is non empty set
C opp is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
~ the Comp of C is Relation-like [: the carrier' of C, the carrier' of C:] -defined the carrier' of C -valued Function-like Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:]
CatStr(# the carrier of C, the carrier' of C, the Target of C, the Source of C,(~ the Comp of C) #) is non empty non void V59() strict CatStr
[:(C opp),C:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of (C opp) is non empty set
[: the carrier of (C opp), the carrier of C:] is Relation-like non empty set
the carrier' of (C opp) is non empty set
[: the carrier' of (C opp), the carrier' of C:] is Relation-like non empty set
the Source of (C opp) is Relation-like the carrier' of (C opp) -defined the carrier of (C opp) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (C opp), the carrier of (C opp):]
[: the carrier' of (C opp), the carrier of (C opp):] is Relation-like non empty set
bool [: the carrier' of (C opp), the carrier of (C opp):] is non empty set
[: the Source of (C opp), the Source of C:] is Relation-like [: the carrier' of (C opp), the carrier' of C:] -defined [: the carrier of (C opp), the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:]
[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:] is Relation-like non empty set
bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:] is non empty set
the Target of (C opp) is Relation-like the carrier' of (C opp) -defined the carrier of (C opp) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (C opp), the carrier of (C opp):]
[: the Target of (C opp), the Target of C:] is Relation-like [: the carrier' of (C opp), the carrier' of C:] -defined [: the carrier of (C opp), the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:]
the Comp of (C opp) is Relation-like [: the carrier' of (C opp), the carrier' of (C opp):] -defined the carrier' of (C opp) -valued Function-like Element of bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):]
[: the carrier' of (C opp), the carrier' of (C opp):] is Relation-like non empty set
[:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):] is Relation-like non empty set
bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):] is non empty set
|: the Comp of (C opp), the Comp of C:| is Relation-like [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:] -defined [: the carrier' of (C opp), the carrier' of C:] -valued Function-like Element of bool [:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:]
[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:] is Relation-like non empty set
[:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:] is Relation-like non empty set
bool [:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:] is non empty set
CatStr(# [: the carrier of (C opp), the carrier of C:],[: the carrier' of (C opp), the carrier' of C:],[: the Source of (C opp), the Source of C:],[: the Target of (C opp), the Target of C:],|: the Comp of (C opp), the Comp of C:| #) is non empty non void V59() strict CatStr
the carrier' of [:(C opp),C:] is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of C is non empty set
the carrier' of C is non empty set
(C) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
bool the carrier' of C is non empty set
[:(bool the carrier' of C),(bool the carrier' of C):] is Relation-like non empty set
a is Element of the carrier of C
(V,C,a) is Relation-like the carrier' of C -defined the carrier' of (V) -valued Function-like non empty total quasi_total Functor of C,(V)
b is Element of the carrier' of C
(V,C,a) . b is Element of the carrier' of (V)
dom b is Element of the carrier of C
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
the Source of C . b is Element of the carrier of C
Hom (a,(dom b)) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = dom b ) } is set
cod b is Element of the carrier of C
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
the Target of C . b is Element of the carrier of C
Hom (a,(cod b)) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = cod b ) } is set
[:(Hom (a,(dom b))),(Hom (a,(cod b))):] is Relation-like set
bool [:(Hom (a,(dom b))),(Hom (a,(cod b))):] is non empty set
[(Hom (a,(dom b))),(Hom (a,(cod b)))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom (a,(dom b))),(Hom (a,(cod b)))} is non empty set
{(Hom (a,(dom b)))} is non empty set
{{(Hom (a,(dom b))),(Hom (a,(cod b)))},{(Hom (a,(dom b)))}} is non empty set
(C,a,b) is Relation-like Hom (a,(dom b)) -defined Hom (a,(cod b)) -valued Function-like quasi_total Element of bool [:(Hom (a,(dom b))),(Hom (a,(cod b))):]
[[(Hom (a,(dom b))),(Hom (a,(cod b)))],(C,a,b)] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom (a,(dom b))),(Hom (a,(cod b))):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom (a,(dom b))),(Hom (a,(cod b))):]):] is Relation-like non empty set
{[(Hom (a,(dom b))),(Hom (a,(cod b)))],(C,a,b)} is non empty set
{[(Hom (a,(dom b))),(Hom (a,(cod b)))]} is Relation-like Function-like non empty set
{{[(Hom (a,(dom b))),(Hom (a,(cod b)))],(C,a,b)},{[(Hom (a,(dom b))),(Hom (a,(cod b)))]}} is non empty set
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(C,a) is Relation-like the carrier' of C -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C,((C)):]
[: the carrier' of C,((C)):] is Relation-like non empty set
bool [: the carrier' of C,((C)):] is non empty set
(C,a) . b is Element of ((C))
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of C is non empty set
(C) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
a is Element of the carrier of C
(V,C,a) is Relation-like the carrier' of C -defined the carrier' of (V) -valued Function-like non empty total quasi_total Functor of C,(V)
the carrier' of C is non empty set
the carrier' of (V) is non empty set
Obj (V,C,a) is Relation-like the carrier of C -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier of C, the carrier of (V):]
[: the carrier of C, the carrier of (V):] is Relation-like non empty set
bool [: the carrier of C, the carrier of (V):] is non empty set
b is Element of the carrier of C
(Obj (V,C,a)) . b is Element of the carrier of (V)
Hom (a,b) is Element of bool the carrier' of C
bool the carrier' of C is non empty set
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = b ) } is set
A is Element of V
(V,A) is Element of the carrier of (V)
id b is Morphism of b,b
(V,C,a) . (id b) is Element of the carrier' of (V)
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(C,a) is Relation-like the carrier' of C -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C,((C)):]
[: the carrier' of C,((C)):] is Relation-like non empty set
bool [: the carrier' of C,((C)):] is non empty set
(C,a) . (id b) is Element of ((C))
id (V,A) is Morphism of (V,A),(V,A)
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of C is non empty set
the carrier' of C is non empty set
(C) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
bool the carrier' of C is non empty set
[:(bool the carrier' of C),(bool the carrier' of C):] is Relation-like non empty set
a is Element of the carrier of C
(V,C,a) is Relation-like the carrier' of C -defined the carrier' of (V) -valued Function-like non empty total quasi_total Contravariant_Functor of C,(V)
b is Element of the carrier' of C
(V,C,a) . b is Element of the carrier' of (V)
cod b is Element of the carrier of C
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
the Target of C . b is Element of the carrier of C
Hom ((cod b),a) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = cod b & cod b₁ = a ) } is set
dom b is Element of the carrier of C
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
the Source of C . b is Element of the carrier of C
Hom ((dom b),a) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = dom b & cod b₁ = a ) } is set
[:(Hom ((cod b),a)),(Hom ((dom b),a)):] is Relation-like set
bool [:(Hom ((cod b),a)),(Hom ((dom b),a)):] is non empty set
[(Hom ((cod b),a)),(Hom ((dom b),a))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom ((cod b),a)),(Hom ((dom b),a))} is non empty set
{(Hom ((cod b),a))} is non empty set
{{(Hom ((cod b),a)),(Hom ((dom b),a))},{(Hom ((cod b),a))}} is non empty set
(C,a,b) is Relation-like Hom ((cod b),a) -defined Hom ((dom b),a) -valued Function-like quasi_total Element of bool [:(Hom ((cod b),a)),(Hom ((dom b),a)):]
[[(Hom ((cod b),a)),(Hom ((dom b),a))],(C,a,b)] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod b),a)),(Hom ((dom b),a)):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod b),a)),(Hom ((dom b),a)):]):] is Relation-like non empty set
{[(Hom ((cod b),a)),(Hom ((dom b),a))],(C,a,b)} is non empty set
{[(Hom ((cod b),a)),(Hom ((dom b),a))]} is Relation-like Function-like non empty set
{{[(Hom ((cod b),a)),(Hom ((dom b),a))],(C,a,b)},{[(Hom ((cod b),a)),(Hom ((dom b),a))]}} is non empty set
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(C,a) is Relation-like the carrier' of C -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C,((C)):]
[: the carrier' of C,((C)):] is Relation-like non empty set
bool [: the carrier' of C,((C)):] is non empty set
(C,a) . b is Element of ((C))
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of C is non empty set
(C) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
a is Element of the carrier of C
(V,C,a) is Relation-like the carrier' of C -defined the carrier' of (V) -valued Function-like non empty total quasi_total Contravariant_Functor of C,(V)
the carrier' of C is non empty set
the carrier' of (V) is non empty set
Obj (V,C,a) is Relation-like the carrier of C -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier of C, the carrier of (V):]
[: the carrier of C, the carrier of (V):] is Relation-like non empty set
bool [: the carrier of C, the carrier of (V):] is non empty set
b is Element of the carrier of C
(Obj (V,C,a)) . b is Element of the carrier of (V)
Hom (b,a) is Element of bool the carrier' of C
bool the carrier' of C is non empty set
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = b & cod b₁ = a ) } is set
A is Element of V
(V,A) is Element of the carrier of (V)
id b is Morphism of b,b
(V,C,a) . (id b) is Element of the carrier' of (V)
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(C,a) is Relation-like the carrier' of C -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C,((C)):]
[: the carrier' of C,((C)):] is Relation-like non empty set
bool [: the carrier' of C,((C)):] is non empty set
(C,a) . (id b) is Element of ((C))
id (V,A) is Morphism of (V,A),(V,A)
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier' of C is non empty set
(C) is non empty set
the carrier of C is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
C opp is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
the Comp of C is Relation-like [: the carrier' of C, the carrier' of C:] -defined the carrier' of C -valued Function-like Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:]
[: the carrier' of C, the carrier' of C:] is Relation-like non empty set
[:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is non empty set
~ the Comp of C is Relation-like [: the carrier' of C, the carrier' of C:] -defined the carrier' of C -valued Function-like Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:]
CatStr(# the carrier of C, the carrier' of C, the Target of C, the Source of C,(~ the Comp of C) #) is non empty non void V59() strict CatStr
[:(C opp),C:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of (C opp) is non empty set
[: the carrier of (C opp), the carrier of C:] is Relation-like non empty set
the carrier' of (C opp) is non empty set
[: the carrier' of (C opp), the carrier' of C:] is Relation-like non empty set
the Source of (C opp) is Relation-like the carrier' of (C opp) -defined the carrier of (C opp) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (C opp), the carrier of (C opp):]
[: the carrier' of (C opp), the carrier of (C opp):] is Relation-like non empty set
bool [: the carrier' of (C opp), the carrier of (C opp):] is non empty set
[: the Source of (C opp), the Source of C:] is Relation-like [: the carrier' of (C opp), the carrier' of C:] -defined [: the carrier of (C opp), the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:]
[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:] is Relation-like non empty set
bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:] is non empty set
the Target of (C opp) is Relation-like the carrier' of (C opp) -defined the carrier of (C opp) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (C opp), the carrier of (C opp):]
[: the Target of (C opp), the Target of C:] is Relation-like [: the carrier' of (C opp), the carrier' of C:] -defined [: the carrier of (C opp), the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:]
the Comp of (C opp) is Relation-like [: the carrier' of (C opp), the carrier' of (C opp):] -defined the carrier' of (C opp) -valued Function-like Element of bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):]
[: the carrier' of (C opp), the carrier' of (C opp):] is Relation-like non empty set
[:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):] is Relation-like non empty set
bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):] is non empty set
|: the Comp of (C opp), the Comp of C:| is Relation-like [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:] -defined [: the carrier' of (C opp), the carrier' of C:] -valued Function-like Element of bool [:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:]
[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:] is Relation-like non empty set
[:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:] is Relation-like non empty set
bool [:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:] is non empty set
CatStr(# [: the carrier of (C opp), the carrier of C:],[: the carrier' of (C opp), the carrier' of C:],[: the Source of (C opp), the Source of C:],[: the Target of (C opp), the Target of C:],|: the Comp of (C opp), the Comp of C:| #) is non empty non void V59() strict CatStr
the carrier' of [:(C opp),C:] is non empty set
(V,C) is Relation-like the carrier' of [:(C opp),C:] -defined the carrier' of (V) -valued Function-like non empty total quasi_total Functor of [:(C opp),C:],(V)
bool the carrier' of C is non empty set
[:(bool the carrier' of C),(bool the carrier' of C):] is Relation-like non empty set
a is Element of the carrier' of C
a opp is Element of the carrier' of (C opp)
cod a is Element of the carrier of C
the Target of C . a is Element of the carrier of C
dom a is Element of the carrier of C
the Source of C . a is Element of the carrier of C
b is Element of the carrier' of C
[(a opp),b] is V15() Element of the carrier' of [:(C opp),C:]
{(a opp),b} is non empty set
{(a opp)} is non empty set
{{(a opp),b},{(a opp)}} is non empty set
(V,C) . [(a opp),b] is Element of the carrier' of (V)
dom b is Element of the carrier of C
the Source of C . b is Element of the carrier of C
Hom ((cod a),(dom b)) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = cod a & cod b₁ = dom b ) } is set
cod b is Element of the carrier of C
the Target of C . b is Element of the carrier of C
Hom ((dom a),(cod b)) is Element of bool the carrier' of C
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = dom a & cod b₁ = cod b ) } is set
[:(Hom ((cod a),(dom b))),(Hom ((dom a),(cod b))):] is Relation-like set
bool [:(Hom ((cod a),(dom b))),(Hom ((dom a),(cod b))):] is non empty set
[(Hom ((cod a),(dom b))),(Hom ((dom a),(cod b)))] is V15() Element of [:(bool the carrier' of C),(bool the carrier' of C):]
{(Hom ((cod a),(dom b))),(Hom ((dom a),(cod b)))} is non empty set
{(Hom ((cod a),(dom b)))} is non empty set
{{(Hom ((cod a),(dom b))),(Hom ((dom a),(cod b)))},{(Hom ((cod a),(dom b)))}} is non empty set
(C,a,b) is Relation-like Hom ((cod a),(dom b)) -defined Hom ((dom a),(cod b)) -valued Function-like quasi_total Element of bool [:(Hom ((cod a),(dom b))),(Hom ((dom a),(cod b))):]
[[(Hom ((cod a),(dom b))),(Hom ((dom a),(cod b)))],(C,a,b)] is V15() Element of [:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod a),(dom b))),(Hom ((dom a),(cod b))):]):]
[:[:(bool the carrier' of C),(bool the carrier' of C):],(bool [:(Hom ((cod a),(dom b))),(Hom ((dom a),(cod b))):]):] is Relation-like non empty set
{[(Hom ((cod a),(dom b))),(Hom ((dom a),(cod b)))],(C,a,b)} is non empty set
{[(Hom ((cod a),(dom b))),(Hom ((dom a),(cod b)))]} is Relation-like Function-like non empty set
{{[(Hom ((cod a),(dom b))),(Hom ((dom a),(cod b)))],(C,a,b)},{[(Hom ((cod a),(dom b))),(Hom ((dom a),(cod b)))]}} is non empty set
[:C,C:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of C, the carrier of C:] is Relation-like non empty set
[: the Source of C, the Source of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is non empty set
[: the Target of C, the Target of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
|: the Comp of C, the Comp of C:| is Relation-like [:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] -defined [: the carrier' of C, the carrier' of C:] -valued Function-like Element of bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
[:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is non empty set
CatStr(# [: the carrier of C, the carrier of C:],[: the carrier' of C, the carrier' of C:],[: the Source of C, the Source of C:],[: the Target of C, the Target of C:],|: the Comp of C, the Comp of C:| #) is non empty non void V59() strict CatStr
the carrier' of [:C,C:] is non empty set
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(C) is Relation-like the carrier' of [:C,C:] -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:C,C:],((C)):]
[: the carrier' of [:C,C:],((C)):] is Relation-like non empty set
bool [: the carrier' of [:C,C:],((C)):] is non empty set
[a,b] is V15() Element of the carrier' of [:C,C:]
{a,b} is non empty set
{a} is non empty set
{{a,b},{a}} is non empty set
(C) . [a,b] is Element of ((C))
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of C is non empty set
(C) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
C opp is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
the carrier' of C is non empty set
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
the Comp of C is Relation-like [: the carrier' of C, the carrier' of C:] -defined the carrier' of C -valued Function-like Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:]
[: the carrier' of C, the carrier' of C:] is Relation-like non empty set
[:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is non empty set
~ the Comp of C is Relation-like [: the carrier' of C, the carrier' of C:] -defined the carrier' of C -valued Function-like Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:]
CatStr(# the carrier of C, the carrier' of C, the Target of C, the Source of C,(~ the Comp of C) #) is non empty non void V59() strict CatStr
[:(C opp),C:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of (C opp) is non empty set
[: the carrier of (C opp), the carrier of C:] is Relation-like non empty set
the carrier' of (C opp) is non empty set
[: the carrier' of (C opp), the carrier' of C:] is Relation-like non empty set
the Source of (C opp) is Relation-like the carrier' of (C opp) -defined the carrier of (C opp) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (C opp), the carrier of (C opp):]
[: the carrier' of (C opp), the carrier of (C opp):] is Relation-like non empty set
bool [: the carrier' of (C opp), the carrier of (C opp):] is non empty set
[: the Source of (C opp), the Source of C:] is Relation-like [: the carrier' of (C opp), the carrier' of C:] -defined [: the carrier of (C opp), the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:]
[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:] is Relation-like non empty set
bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:] is non empty set
the Target of (C opp) is Relation-like the carrier' of (C opp) -defined the carrier of (C opp) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (C opp), the carrier of (C opp):]
[: the Target of (C opp), the Target of C:] is Relation-like [: the carrier' of (C opp), the carrier' of C:] -defined [: the carrier of (C opp), the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:]
the Comp of (C opp) is Relation-like [: the carrier' of (C opp), the carrier' of (C opp):] -defined the carrier' of (C opp) -valued Function-like Element of bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):]
[: the carrier' of (C opp), the carrier' of (C opp):] is Relation-like non empty set
[:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):] is Relation-like non empty set
bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):] is non empty set
|: the Comp of (C opp), the Comp of C:| is Relation-like [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:] -defined [: the carrier' of (C opp), the carrier' of C:] -valued Function-like Element of bool [:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:]
[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:] is Relation-like non empty set
[:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:] is Relation-like non empty set
bool [:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:] is non empty set
CatStr(# [: the carrier of (C opp), the carrier of C:],[: the carrier' of (C opp), the carrier' of C:],[: the Source of (C opp), the Source of C:],[: the Target of (C opp), the Target of C:],|: the Comp of (C opp), the Comp of C:| #) is non empty non void V59() strict CatStr
the carrier of [:(C opp),C:] is non empty set
(V,C) is Relation-like the carrier' of [:(C opp),C:] -defined the carrier' of (V) -valued Function-like non empty total quasi_total Functor of [:(C opp),C:],(V)
the carrier' of [:(C opp),C:] is non empty set
the carrier' of (V) is non empty set
Obj (V,C) is Relation-like the carrier of [:(C opp),C:] -defined the carrier of (V) -valued Function-like non empty total quasi_total Element of bool [: the carrier of [:(C opp),C:], the carrier of (V):]
[: the carrier of [:(C opp),C:], the carrier of (V):] is Relation-like non empty set
bool [: the carrier of [:(C opp),C:], the carrier of (V):] is non empty set
a is Element of the carrier of C
a opp is Element of the carrier of (C opp)
b is Element of the carrier of C
[(a opp),b] is V15() Element of the carrier of [:(C opp),C:]
{(a opp),b} is non empty set
{(a opp)} is non empty set
{{(a opp),b},{(a opp)}} is non empty set
(Obj (V,C)) . [(a opp),b] is Element of the carrier of (V)
Hom (a,b) is Element of bool the carrier' of C
bool the carrier' of C is non empty set
{ b₁ where b₁ is Element of the carrier' of C : ( dom b₁ = a & cod b₁ = b ) } is set
id (a opp) is Morphism of a opp ,a opp
id a is Morphism of a,a
A is Element of V
(V,A) is Element of the carrier of (V)
id [(a opp),b] is Morphism of [(a opp),b],[(a opp),b]
(V,C) . (id [(a opp),b]) is Element of the carrier' of (V)
id b is Morphism of b,b
[(id (a opp)),(id b)] is V15() Element of the carrier' of [:(C opp),C:]
{(id (a opp)),(id b)} is non empty set
{(id (a opp))} is non empty set
{{(id (a opp)),(id b)},{(id (a opp))}} is non empty set
(V,C) . [(id (a opp)),(id b)] is Element of the carrier' of (V)
[:C,C:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of C, the carrier of C:] is Relation-like non empty set
[: the Source of C, the Source of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is non empty set
[: the Target of C, the Target of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
|: the Comp of C, the Comp of C:| is Relation-like [:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] -defined [: the carrier' of C, the carrier' of C:] -valued Function-like Element of bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
[:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is non empty set
CatStr(# [: the carrier of C, the carrier of C:],[: the carrier' of C, the carrier' of C:],[: the Source of C, the Source of C:],[: the Target of C, the Target of C:],|: the Comp of C, the Comp of C:| #) is non empty non void V59() strict CatStr
the carrier' of [:C,C:] is non empty set
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(C) is Relation-like the carrier' of [:C,C:] -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:C,C:],((C)):]
[: the carrier' of [:C,C:],((C)):] is Relation-like non empty set
bool [: the carrier' of [:C,C:],((C)):] is non empty set
[(id a),(id b)] is V15() Element of the carrier' of [:C,C:]
{(id a),(id b)} is non empty set
{(id a)} is non empty set
{{(id a),(id b)},{(id a)}} is non empty set
(C) . [(id a),(id b)] is Element of ((C))
id (V,A) is Morphism of (V,A),(V,A)
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of C is non empty set
(C) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
the carrier' of C is non empty set
C opp is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
the Comp of C is Relation-like [: the carrier' of C, the carrier' of C:] -defined the carrier' of C -valued Function-like Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:]
[: the carrier' of C, the carrier' of C:] is Relation-like non empty set
[:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is non empty set
~ the Comp of C is Relation-like [: the carrier' of C, the carrier' of C:] -defined the carrier' of C -valued Function-like Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:]
CatStr(# the carrier of C, the carrier' of C, the Target of C, the Source of C,(~ the Comp of C) #) is non empty non void V59() strict CatStr
(V,C) is Relation-like the carrier' of [:(C opp),C:] -defined the carrier' of (V) -valued Function-like non empty total quasi_total Functor of [:(C opp),C:],(V)
[:(C opp),C:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of (C opp) is non empty set
[: the carrier of (C opp), the carrier of C:] is Relation-like non empty set
the carrier' of (C opp) is non empty set
[: the carrier' of (C opp), the carrier' of C:] is Relation-like non empty set
the Source of (C opp) is Relation-like the carrier' of (C opp) -defined the carrier of (C opp) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (C opp), the carrier of (C opp):]
[: the carrier' of (C opp), the carrier of (C opp):] is Relation-like non empty set
bool [: the carrier' of (C opp), the carrier of (C opp):] is non empty set
[: the Source of (C opp), the Source of C:] is Relation-like [: the carrier' of (C opp), the carrier' of C:] -defined [: the carrier of (C opp), the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:]
[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:] is Relation-like non empty set
bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:] is non empty set
the Target of (C opp) is Relation-like the carrier' of (C opp) -defined the carrier of (C opp) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (C opp), the carrier of (C opp):]
[: the Target of (C opp), the Target of C:] is Relation-like [: the carrier' of (C opp), the carrier' of C:] -defined [: the carrier of (C opp), the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:]
the Comp of (C opp) is Relation-like [: the carrier' of (C opp), the carrier' of (C opp):] -defined the carrier' of (C opp) -valued Function-like Element of bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):]
[: the carrier' of (C opp), the carrier' of (C opp):] is Relation-like non empty set
[:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):] is Relation-like non empty set
bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):] is non empty set
|: the Comp of (C opp), the Comp of C:| is Relation-like [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:] -defined [: the carrier' of (C opp), the carrier' of C:] -valued Function-like Element of bool [:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:]
[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:] is Relation-like non empty set
[:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:] is Relation-like non empty set
bool [:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:] is non empty set
CatStr(# [: the carrier of (C opp), the carrier of C:],[: the carrier' of (C opp), the carrier' of C:],[: the Source of (C opp), the Source of C:],[: the Target of (C opp), the Target of C:],|: the Comp of (C opp), the Comp of C:| #) is non empty non void V59() strict CatStr
the carrier' of [:(C opp),C:] is non empty set
a is Element of the carrier of C
a opp is Element of the carrier of (C opp)
(V,C) ?- (a opp) is Relation-like the carrier' of C -defined the carrier' of (V) -valued Function-like non empty total quasi_total Functor of C,(V)
curry (V,C) is Relation-like Function-like set
id (a opp) is Morphism of a opp ,a opp
(curry (V,C)) . (id (a opp)) is set
(V,C,a) is Relation-like the carrier' of C -defined the carrier' of (V) -valued Function-like non empty total quasi_total Functor of C,(V)
id a is Morphism of a,a
(C) is Relation-like the carrier' of [:C,C:] -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:C,C:],((C)):]
[:C,C:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of C, the carrier of C:] is Relation-like non empty set
[: the Source of C, the Source of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is non empty set
[: the Target of C, the Target of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
|: the Comp of C, the Comp of C:| is Relation-like [:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] -defined [: the carrier' of C, the carrier' of C:] -valued Function-like Element of bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
[:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is non empty set
CatStr(# [: the carrier of C, the carrier of C:],[: the carrier' of C, the carrier' of C:],[: the Source of C, the Source of C:],[: the Target of C, the Target of C:],|: the Comp of C, the Comp of C:| #) is non empty non void V59() strict CatStr
the carrier' of [:C,C:] is non empty set
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
[: the carrier' of [:C,C:],((C)):] is Relation-like non empty set
bool [: the carrier' of [:C,C:],((C)):] is non empty set
curry (C) is Relation-like Function-like set
(curry (C)) . (id a) is set
(C,a) is Relation-like the carrier' of C -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C,((C)):]
[: the carrier' of C,((C)):] is Relation-like non empty set
bool [: the carrier' of C,((C)):] is non empty set
V is non empty set
(V) is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
(V) is non empty set
(V) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of V : verum } is set
[:V,V:] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of V, b₃ is Relation-like Function-like Element of (V) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
[:(V),V:] is Relation-like non empty set
bool [:(V),V:] is non empty set
(V) is Relation-like (V) -defined V -valued Function-like non empty total quasi_total Element of bool [:(V),V:]
(V) is Relation-like [:(V),(V):] -defined (V) -valued Function-like Element of bool [:[:(V),(V):],(V):]
[:(V),(V):] is Relation-like non empty set
[:[:(V),(V):],(V):] is Relation-like non empty set
bool [:[:(V),(V):],(V):] is non empty set
CatStr(# V,(V),(V),(V),(V) #) is non empty non void V59() strict CatStr
the carrier' of (V) is non empty set
C is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of C is non empty set
(C) is non empty set
{ (Hom (b₁,b₂)) where b₁, b₂ is Element of the carrier of C : verum } is set
C opp is non empty non void V59() strict Category-like transitive associative reflexive with_identities CatStr
the carrier' of C is non empty set
the Target of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
[: the carrier' of C, the carrier of C:] is Relation-like non empty set
bool [: the carrier' of C, the carrier of C:] is non empty set
the Source of C is Relation-like the carrier' of C -defined the carrier of C -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C, the carrier of C:]
the Comp of C is Relation-like [: the carrier' of C, the carrier' of C:] -defined the carrier' of C -valued Function-like Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:]
[: the carrier' of C, the carrier' of C:] is Relation-like non empty set
[:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:] is non empty set
~ the Comp of C is Relation-like [: the carrier' of C, the carrier' of C:] -defined the carrier' of C -valued Function-like Element of bool [:[: the carrier' of C, the carrier' of C:], the carrier' of C:]
CatStr(# the carrier of C, the carrier' of C, the Target of C, the Source of C,(~ the Comp of C) #) is non empty non void V59() strict CatStr
the carrier' of (C opp) is non empty set
(V,C) is Relation-like the carrier' of [:(C opp),C:] -defined the carrier' of (V) -valued Function-like non empty total quasi_total Functor of [:(C opp),C:],(V)
[:(C opp),C:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
the carrier of (C opp) is non empty set
[: the carrier of (C opp), the carrier of C:] is Relation-like non empty set
[: the carrier' of (C opp), the carrier' of C:] is Relation-like non empty set
the Source of (C opp) is Relation-like the carrier' of (C opp) -defined the carrier of (C opp) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (C opp), the carrier of (C opp):]
[: the carrier' of (C opp), the carrier of (C opp):] is Relation-like non empty set
bool [: the carrier' of (C opp), the carrier of (C opp):] is non empty set
[: the Source of (C opp), the Source of C:] is Relation-like [: the carrier' of (C opp), the carrier' of C:] -defined [: the carrier of (C opp), the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:]
[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:] is Relation-like non empty set
bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:] is non empty set
the Target of (C opp) is Relation-like the carrier' of (C opp) -defined the carrier of (C opp) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of (C opp), the carrier of (C opp):]
[: the Target of (C opp), the Target of C:] is Relation-like [: the carrier' of (C opp), the carrier' of C:] -defined [: the carrier of (C opp), the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier of (C opp), the carrier of C:]:]
the Comp of (C opp) is Relation-like [: the carrier' of (C opp), the carrier' of (C opp):] -defined the carrier' of (C opp) -valued Function-like Element of bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):]
[: the carrier' of (C opp), the carrier' of (C opp):] is Relation-like non empty set
[:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):] is Relation-like non empty set
bool [:[: the carrier' of (C opp), the carrier' of (C opp):], the carrier' of (C opp):] is non empty set
|: the Comp of (C opp), the Comp of C:| is Relation-like [:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:] -defined [: the carrier' of (C opp), the carrier' of C:] -valued Function-like Element of bool [:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:]
[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:] is Relation-like non empty set
[:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:] is Relation-like non empty set
bool [:[:[: the carrier' of (C opp), the carrier' of C:],[: the carrier' of (C opp), the carrier' of C:]:],[: the carrier' of (C opp), the carrier' of C:]:] is non empty set
CatStr(# [: the carrier of (C opp), the carrier of C:],[: the carrier' of (C opp), the carrier' of C:],[: the Source of (C opp), the Source of C:],[: the Target of (C opp), the Target of C:],|: the Comp of (C opp), the Comp of C:| #) is non empty non void V59() strict CatStr
the carrier' of [:(C opp),C:] is non empty set
a is Element of the carrier of C
(V,C) -? a is Relation-like the carrier' of (C opp) -defined the carrier' of (V) -valued Function-like non empty total quasi_total Functor of C opp ,(V)
curry' (V,C) is Relation-like Function-like set
id a is Morphism of a,a
(curry' (V,C)) . (id a) is set
(V,C,a) is Relation-like the carrier' of C -defined the carrier' of (V) -valued Function-like non empty total quasi_total Contravariant_Functor of C,(V)
(C) is Relation-like the carrier' of [:C,C:] -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of [:C,C:],((C)):]
[:C,C:] is non empty non void V59() Category-like transitive associative reflexive with_identities CatStr
[: the carrier of C, the carrier of C:] is Relation-like non empty set
[: the Source of C, the Source of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is Relation-like non empty set
bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:] is non empty set
[: the Target of C, the Target of C:] is Relation-like [: the carrier' of C, the carrier' of C:] -defined [: the carrier of C, the carrier of C:] -valued Function-like non empty total quasi_total Element of bool [:[: the carrier' of C, the carrier' of C:],[: the carrier of C, the carrier of C:]:]
|: the Comp of C, the Comp of C:| is Relation-like [:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] -defined [: the carrier' of C, the carrier' of C:] -valued Function-like Element of bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:]
[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
[:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is Relation-like non empty set
bool [:[:[: the carrier' of C, the carrier' of C:],[: the carrier' of C, the carrier' of C:]:],[: the carrier' of C, the carrier' of C:]:] is non empty set
CatStr(# [: the carrier of C, the carrier of C:],[: the carrier' of C, the carrier' of C:],[: the Source of C, the Source of C:],[: the Target of C, the Target of C:],|: the Comp of C, the Comp of C:| #) is non empty non void V59() strict CatStr
the carrier' of [:C,C:] is non empty set
((C)) is non empty set
((C)) is functional non empty set
{ (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
union { (Funcs (b₁,b₂)) where b₁, b₂ is Element of (C) : verum } is set
[:(C),(C):] is Relation-like non empty set
{ [[b₁,b₂],b₃] where b₁, b₂ is Element of (C), b₃ is Relation-like Function-like Element of ((C)) : ( ( not b₂ = {} or b₁ = {} ) & b₃ is Relation-like b₁ -defined b₂ -valued Function-like quasi_total Element of bool [:b₁,b₂:] ) } is set
[: the carrier' of [:C,C:],((C)):] is Relation-like non empty set
bool [: the carrier' of [:C,C:],((C)):] is non empty set
curry' (C) is Relation-like Function-like set
(curry' (C)) . (id a) is set
(C,a) is Relation-like the carrier' of C -defined ((C)) -valued Function-like non empty total quasi_total Element of bool [: the carrier' of C,((C)):]
[: the carrier' of C,((C)):] is Relation-like non empty set
bool [: the carrier' of C,((C)):] is non empty set