:: CARD_FIN semantic presentation
begin
theorem Th1: :: CARD_FIN:1
for X, Y being finite set st X c= Y & card X = card Y holds
X = Y
proof
let X, Y be finite set ; ::_thesis: ( X c= Y & card X = card Y implies X = Y )
assume that
A1: X c= Y and
A2: card X = card Y ; ::_thesis: X = Y
card (Y \ X) = (card Y) - (card X) by A1, CARD_2:44;
then Y \ X = {} by A2;
then Y c= X by XBOOLE_1:37;
hence X = Y by A1, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th2: :: CARD_FIN:2
for X, Y being finite set
for x, y being set st ( Y = {} implies X = {} ) & not x in X holds
card (Funcs (X,Y)) = card { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) }
proof
defpred S1[ set , set , set ] means 1 = 1;
let X, Y be finite set ; ::_thesis: for x, y being set st ( Y = {} implies X = {} ) & not x in X holds
card (Funcs (X,Y)) = card { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) }
let x, y be set ; ::_thesis: ( ( Y = {} implies X = {} ) & not x in X implies card (Funcs (X,Y)) = card { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } )
assume A1: ( Y = {} implies X = {} ) ; ::_thesis: ( x in X or card (Funcs (X,Y)) = card { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } )
set F2 = { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) } ;
A2: for f being Function of (X \/ {x}),(Y \/ {y}) st f . x = y holds
( S1[f,X \/ {x},Y \/ {y}] iff S1[f | X,X,Y] ) ;
set F1 = { f where f is Function of X,Y : S1[f,X,Y] } ;
assume A3: not x in X ; ::_thesis: card (Funcs (X,Y)) = card { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) }
set F3 = { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } ;
A4: Funcs (X,Y) c= { f where f is Function of X,Y : S1[f,X,Y] }
proof
let F be set ; :: according to TARSKI:def_3 ::_thesis: ( not F in Funcs (X,Y) or F in { f where f is Function of X,Y : S1[f,X,Y] } )
assume F in Funcs (X,Y) ; ::_thesis: F in { f where f is Function of X,Y : S1[f,X,Y] }
then F is Function of X,Y by FUNCT_2:66;
hence F in { f where f is Function of X,Y : S1[f,X,Y] } ; ::_thesis: verum
end;
A5: { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) } c= { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) }
proof
let F be set ; :: according to TARSKI:def_3 ::_thesis: ( not F in { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) } or F in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } )
assume F in { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) } ; ::_thesis: F in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) }
then ex f being Function of (X \/ {x}),(Y \/ {y}) st
( f = F & S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) ;
hence F in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } ; ::_thesis: verum
end;
A6: ( Y is empty implies X is empty ) by A1;
A7: card { f where f is Function of X,Y : S1[f,X,Y] } = card { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) } from STIRL2_1:sch_4(A6, A3, A2);
A8: { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } c= { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) }
proof
let F be set ; :: according to TARSKI:def_3 ::_thesis: ( not F in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } or F in { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) } )
assume F in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } ; ::_thesis: F in { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) }
then ex f being Function of (X \/ {x}),(Y \/ {y}) st
( f = F & rng (f | X) c= Y & f . x = y ) ;
hence F in { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) } ; ::_thesis: verum
end;
{ f where f is Function of X,Y : S1[f,X,Y] } c= Funcs (X,Y)
proof
let F be set ; :: according to TARSKI:def_3 ::_thesis: ( not F in { f where f is Function of X,Y : S1[f,X,Y] } or F in Funcs (X,Y) )
assume F in { f where f is Function of X,Y : S1[f,X,Y] } ; ::_thesis: F in Funcs (X,Y)
then ex f being Function of X,Y st
( f = F & S1[f,X,Y] ) ;
hence F in Funcs (X,Y) by A1, FUNCT_2:8; ::_thesis: verum
end;
then Funcs (X,Y) = { f where f is Function of X,Y : S1[f,X,Y] } by A4, XBOOLE_0:def_10;
hence card (Funcs (X,Y)) = card { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } by A5, A8, A7, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th3: :: CARD_FIN:3
for X, Y being finite set
for x, y being set st not x in X & y in Y holds
card (Funcs (X,Y)) = card { F where F is Function of (X \/ {x}),Y : F . x = y }
proof
let X, Y be finite set ; ::_thesis: for x, y being set st not x in X & y in Y holds
card (Funcs (X,Y)) = card { F where F is Function of (X \/ {x}),Y : F . x = y }
let x, y be set ; ::_thesis: ( not x in X & y in Y implies card (Funcs (X,Y)) = card { F where F is Function of (X \/ {x}),Y : F . x = y } )
assume that
A1: not x in X and
A2: y in Y ; ::_thesis: card (Funcs (X,Y)) = card { F where F is Function of (X \/ {x}),Y : F . x = y }
set F2 = { F where F is Function of (X \/ {x}),Y : F . x = y } ;
set F1 = { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } ;
{y} c= Y by A2, ZFMISC_1:31;
then A3: Y = Y \/ {y} by XBOOLE_1:12;
A4: { F where F is Function of (X \/ {x}),Y : F . x = y } c= { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) }
proof
let f be set ; :: according to TARSKI:def_3 ::_thesis: ( not f in { F where F is Function of (X \/ {x}),Y : F . x = y } or f in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } )
assume f in { F where F is Function of (X \/ {x}),Y : F . x = y } ; ::_thesis: f in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) }
then consider F being Function of (X \/ {x}),Y such that
A5: ( f = F & F . x = y ) ;
rng (F | X) c= Y ;
hence f in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } by A3, A5; ::_thesis: verum
end;
{ F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } c= { F where F is Function of (X \/ {x}),Y : F . x = y }
proof
let f be set ; :: according to TARSKI:def_3 ::_thesis: ( not f in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } or f in { F where F is Function of (X \/ {x}),Y : F . x = y } )
assume f in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } ; ::_thesis: f in { F where F is Function of (X \/ {x}),Y : F . x = y }
then ex F being Function of (X \/ {x}),(Y \/ {y}) st
( f = F & rng (F | X) c= Y & F . x = y ) ;
hence f in { F where F is Function of (X \/ {x}),Y : F . x = y } by A3; ::_thesis: verum
end;
then { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( rng (F | X) c= Y & F . x = y ) } = { F where F is Function of (X \/ {x}),Y : F . x = y } by A4, XBOOLE_0:def_10;
hence card (Funcs (X,Y)) = card { F where F is Function of (X \/ {x}),Y : F . x = y } by A1, A2, Th2; ::_thesis: verum
end;
theorem Th4: :: CARD_FIN:4
for Y, X being finite set st ( Y = {} implies X = {} ) holds
card (Funcs (X,Y)) = (card Y) |^ (card X)
proof
let Y, X be finite set ; ::_thesis: ( ( Y = {} implies X = {} ) implies card (Funcs (X,Y)) = (card Y) |^ (card X) )
assume A1: ( Y = {} implies X = {} ) ; ::_thesis: card (Funcs (X,Y)) = (card Y) |^ (card X)
percases ( Y is empty or not Y is empty ) ;
supposeA2: Y is empty ; ::_thesis: card (Funcs (X,Y)) = (card Y) |^ (card X)
then card (Funcs (X,Y)) = 1 by A1, ALTCAT_1:2, CARD_1:30;
hence card (Funcs (X,Y)) = (card Y) |^ (card X) by A1, A2, CARD_1:27, NEWTON:4; ::_thesis: verum
end;
supposeA3: not Y is empty ; ::_thesis: card (Funcs (X,Y)) = (card Y) |^ (card X)
defpred S1[ Nat] means for X, Y being finite set st not Y is empty & card X = $1 holds
card (Funcs (X,Y)) = (card Y) |^ (card X);
A4: for n being Nat st S1[n] holds
S1[n + 1]
proof
defpred S2[ set ] means 1 = 1;
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A5: S1[n] ; ::_thesis: S1[n + 1]
let X, Y be finite set ; ::_thesis: ( not Y is empty & card X = n + 1 implies card (Funcs (X,Y)) = (card Y) |^ (card X) )
assume that
A6: not Y is empty and
A7: card X = n + 1 ; ::_thesis: card (Funcs (X,Y)) = (card Y) |^ (card X)
reconsider nn = n as Element of NAT by ORDINAL1:def_12;
reconsider cY = (card Y) |^ nn as Element of NAT ;
card Y,Y are_equipotent by CARD_1:def_2;
then consider f being Function such that
A8: f is one-to-one and
A9: dom f = card Y and
A10: rng f = Y by WELLORD2:def_4;
reconsider f = f as Function of (card Y),Y by A9, A10, FUNCT_2:1;
consider x being set such that
A11: x in X by A7, CARD_1:27, XBOOLE_0:def_1;
A12: ( f is onto & f is one-to-one ) by A8, A10, FUNCT_2:def_3;
consider F being XFinSequence of such that
A13: dom F = card Y and
A14: card { g where g is Function of X,Y : S2[g] } = Sum F and
A15: for k being Nat st k in dom F holds
F . k = card { g where g is Function of X,Y : ( S2[g] & g . x = f . k ) } from STIRL2_1:sch_6(A12, A6, A11);
A16: for k being Nat st k in dom F holds
F . k = cY
proof
set Xx = X \ {x};
let k be Nat; ::_thesis: ( k in dom F implies F . k = cY )
assume A17: k in dom F ; ::_thesis: F . k = cY
A18: f . k in rng f by A9, A13, A17, FUNCT_1:def_3;
set F3 = { g where g is Function of X,Y : ( S2[g] & g . x = f . k ) } ;
set F2 = { g where g is Function of ((X \ {x}) \/ {x}),Y : g . x = f . k } ;
A19: { g where g is Function of X,Y : ( S2[g] & g . x = f . k ) } c= { g where g is Function of ((X \ {x}) \/ {x}),Y : g . x = f . k }
proof
let G be set ; :: according to TARSKI:def_3 ::_thesis: ( not G in { g where g is Function of X,Y : ( S2[g] & g . x = f . k ) } or G in { g where g is Function of ((X \ {x}) \/ {x}),Y : g . x = f . k } )
assume G in { g where g is Function of X,Y : ( S2[g] & g . x = f . k ) } ; ::_thesis: G in { g where g is Function of ((X \ {x}) \/ {x}),Y : g . x = f . k }
then A20: ex g being Function of X,Y st
( g = G & S2[g] & g . x = f . k ) ;
(X \ {x}) \/ {x} = X by A11, ZFMISC_1:116;
hence G in { g where g is Function of ((X \ {x}) \/ {x}),Y : g . x = f . k } by A20; ::_thesis: verum
end;
{ g where g is Function of ((X \ {x}) \/ {x}),Y : g . x = f . k } c= { g where g is Function of X,Y : ( S2[g] & g . x = f . k ) }
proof
let G be set ; :: according to TARSKI:def_3 ::_thesis: ( not G in { g where g is Function of ((X \ {x}) \/ {x}),Y : g . x = f . k } or G in { g where g is Function of X,Y : ( S2[g] & g . x = f . k ) } )
assume G in { g where g is Function of ((X \ {x}) \/ {x}),Y : g . x = f . k } ; ::_thesis: G in { g where g is Function of X,Y : ( S2[g] & g . x = f . k ) }
then A21: ex g being Function of ((X \ {x}) \/ {x}),Y st
( g = G & g . x = f . k ) ;
(X \ {x}) \/ {x} = X by A11, ZFMISC_1:116;
hence G in { g where g is Function of X,Y : ( S2[g] & g . x = f . k ) } by A21; ::_thesis: verum
end;
then A22: { g where g is Function of ((X \ {x}) \/ {x}),Y : g . x = f . k } = { g where g is Function of X,Y : ( S2[g] & g . x = f . k ) } by A19, XBOOLE_0:def_10;
card (X \ {x}) = n by A7, A11, STIRL2_1:55;
then A23: card (Funcs ((X \ {x}),Y)) = cY by A5, A18;
x in {x} by TARSKI:def_1;
then not x in X \ {x} by XBOOLE_0:def_5;
then card (Funcs ((X \ {x}),Y)) = card { g where g is Function of ((X \ {x}) \/ {x}),Y : g . x = f . k } by A18, Th3;
hence F . k = cY by A15, A17, A22, A23; ::_thesis: verum
end;
then for k being Nat st k in dom F holds
F . k >= cY ;
then A24: Sum F >= (len F) * ((card Y) |^ n) by AFINSQ_2:60;
set F1 = { g where g is Function of X,Y : S2[g] } ;
A25: Funcs (X,Y) c= { g where g is Function of X,Y : S2[g] }
proof
let G be set ; :: according to TARSKI:def_3 ::_thesis: ( not G in Funcs (X,Y) or G in { g where g is Function of X,Y : S2[g] } )
assume G in Funcs (X,Y) ; ::_thesis: G in { g where g is Function of X,Y : S2[g] }
then G is Function of X,Y by FUNCT_2:66;
hence G in { g where g is Function of X,Y : S2[g] } ; ::_thesis: verum
end;
{ g where g is Function of X,Y : S2[g] } c= Funcs (X,Y)
proof
let G be set ; :: according to TARSKI:def_3 ::_thesis: ( not G in { g where g is Function of X,Y : S2[g] } or G in Funcs (X,Y) )
assume G in { g where g is Function of X,Y : S2[g] } ; ::_thesis: G in Funcs (X,Y)
then ex g being Function of X,Y st
( g = G & S2[g] ) ;
hence G in Funcs (X,Y) by A6, FUNCT_2:8; ::_thesis: verum
end;
then A26: Funcs (X,Y) = { g where g is Function of X,Y : S2[g] } by A25, XBOOLE_0:def_10;
for k being Nat st k in dom F holds
F . k <= cY by A16;
then Sum F <= (len F) * ((card Y) |^ n) by AFINSQ_2:59;
then Sum F = (card Y) * ((card Y) |^ n) by A13, A24, XXREAL_0:1;
hence card (Funcs (X,Y)) = (card Y) |^ (card X) by A7, A14, A26, NEWTON:6; ::_thesis: verum
end;
A27: S1[ 0 ]
proof
let X, Y be finite set ; ::_thesis: ( not Y is empty & card X = 0 implies card (Funcs (X,Y)) = (card Y) |^ (card X) )
assume that
not Y is empty and
A28: card X = 0 ; ::_thesis: card (Funcs (X,Y)) = (card Y) |^ (card X)
X is empty by A28;
then Funcs (X,Y) = {{}} by FUNCT_5:57;
then card (Funcs (X,Y)) = 1 by CARD_1:30;
hence card (Funcs (X,Y)) = (card Y) |^ (card X) by A28, NEWTON:4; ::_thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch_2(A27, A4);
hence card (Funcs (X,Y)) = (card Y) |^ (card X) by A3; ::_thesis: verum
end;
end;
end;
theorem Th5: :: CARD_FIN:5
for X, Y being finite set
for x, y being set st ( Y is empty implies X is empty ) & not x in X & not y in Y holds
card { F where F is Function of X,Y : F is one-to-one } = card { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( F is one-to-one & F . x = y ) }
proof
let X, Y be finite set ; ::_thesis: for x, y being set st ( Y is empty implies X is empty ) & not x in X & not y in Y holds
card { F where F is Function of X,Y : F is one-to-one } = card { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( F is one-to-one & F . x = y ) }
let x, y be set ; ::_thesis: ( ( Y is empty implies X is empty ) & not x in X & not y in Y implies card { F where F is Function of X,Y : F is one-to-one } = card { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( F is one-to-one & F . x = y ) } )
assume that
A1: ( Y is empty implies X is empty ) and
A2: not x in X and
A3: not y in Y ; ::_thesis: card { F where F is Function of X,Y : F is one-to-one } = card { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( F is one-to-one & F . x = y ) }
defpred S1[ Function, set , set ] means ( $1 is one-to-one & rng ($1 | X) c= Y );
A4: for f being Function of (X \/ {x}),(Y \/ {y}) st f . x = y holds
( S1[f,X \/ {x},Y \/ {y}] iff S1[f | X,X,Y] )
proof
let f be Function of (X \/ {x}),(Y \/ {y}); ::_thesis: ( f . x = y implies ( S1[f,X \/ {x},Y \/ {y}] iff S1[f | X,X,Y] ) )
assume A5: f . x = y ; ::_thesis: ( S1[f,X \/ {x},Y \/ {y}] iff S1[f | X,X,Y] )
thus ( S1[f,X \/ {x},Y \/ {y}] implies S1[f | X,X,Y] ) by FUNCT_1:52; ::_thesis: ( S1[f | X,X,Y] implies S1[f,X \/ {x},Y \/ {y}] )
thus ( S1[f | X,X,Y] implies S1[f,X \/ {x},Y \/ {y}] ) ::_thesis: verum
proof
( (X \/ {x}) /\ X = X & dom f = X \/ {x} ) by FUNCT_2:def_1, XBOOLE_1:21;
then A6: dom (f | X) = X by RELAT_1:61;
assume that
A7: f | X is one-to-one and
A8: rng ((f | X) | X) c= Y ; ::_thesis: S1[f,X \/ {x},Y \/ {y}]
rng (f | X) c= Y by A8;
then f | X is Function of X,Y by A6, FUNCT_2:2;
hence S1[f,X \/ {x},Y \/ {y}] by A1, A3, A5, A7, A8, STIRL2_1:58; ::_thesis: verum
end;
end;
set F3 = { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( F is one-to-one & F . x = y ) } ;
A9: { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( F is one-to-one & F . x = y ) } c= { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) }
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( F is one-to-one & F . x = y ) } or z in { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) } )
assume z in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( F is one-to-one & F . x = y ) } ; ::_thesis: z in { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) }
then consider F being Function of (X \/ {x}),(Y \/ {y}) such that
A10: z = F and
A11: ( F is one-to-one & F . x = y ) ;
rng (F | X) c= Y
proof
A12: dom F = X \/ {x} by FUNCT_2:def_1;
x in {x} by TARSKI:def_1;
then A13: x in dom F by A12, XBOOLE_0:def_3;
assume not rng (F | X) c= Y ; ::_thesis: contradiction
then consider fz being set such that
A14: fz in rng (F | X) and
A15: not fz in Y by TARSKI:def_3;
consider z being set such that
A16: z in dom (F | X) and
A17: fz = (F | X) . z by A14, FUNCT_1:def_3;
A18: z in dom F by A16, RELAT_1:57;
A19: ( fz in Y or fz in {y} ) by A14, XBOOLE_0:def_3;
A20: z in X by A16;
F . z = (F | X) . z by A16, FUNCT_1:47;
then y = F . z by A15, A17, A19, TARSKI:def_1;
hence contradiction by A2, A11, A13, A20, A18, FUNCT_1:def_4; ::_thesis: verum
end;
hence z in { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) } by A10, A11; ::_thesis: verum
end;
A21: { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) } c= { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( F is one-to-one & F . x = y ) }
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) } or z in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( F is one-to-one & F . x = y ) } )
assume z in { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) } ; ::_thesis: z in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( F is one-to-one & F . x = y ) }
then ex f being Function of (X \/ {x}),(Y \/ {y}) st
( z = f & S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) ;
hence z in { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( F is one-to-one & F . x = y ) } ; ::_thesis: verum
end;
set F2 = { f where f is Function of X,Y : ( f is one-to-one & rng (f | X) c= Y ) } ;
set F1 = { F where F is Function of X,Y : F is one-to-one } ;
A22: { F where F is Function of X,Y : F is one-to-one } c= { f where f is Function of X,Y : ( f is one-to-one & rng (f | X) c= Y ) }
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { F where F is Function of X,Y : F is one-to-one } or z in { f where f is Function of X,Y : ( f is one-to-one & rng (f | X) c= Y ) } )
assume z in { F where F is Function of X,Y : F is one-to-one } ; ::_thesis: z in { f where f is Function of X,Y : ( f is one-to-one & rng (f | X) c= Y ) }
then consider F being Function of X,Y such that
A23: ( z = F & F is one-to-one ) ;
rng (F | X) c= rng F ;
hence z in { f where f is Function of X,Y : ( f is one-to-one & rng (f | X) c= Y ) } by A23; ::_thesis: verum
end;
A24: not x in X by A2;
A25: card { f where f is Function of X,Y : S1[f,X,Y] } = card { f where f is Function of (X \/ {x}),(Y \/ {y}) : ( S1[f,X \/ {x},Y \/ {y}] & rng (f | X) c= Y & f . x = y ) } from STIRL2_1:sch_4(A1, A24, A4);
{ f where f is Function of X,Y : ( f is one-to-one & rng (f | X) c= Y ) } c= { F where F is Function of X,Y : F is one-to-one }
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in { f where f is Function of X,Y : ( f is one-to-one & rng (f | X) c= Y ) } or z in { F where F is Function of X,Y : F is one-to-one } )
assume z in { f where f is Function of X,Y : ( f is one-to-one & rng (f | X) c= Y ) } ; ::_thesis: z in { F where F is Function of X,Y : F is one-to-one }
then ex f being Function of X,Y st
( z = f & f is one-to-one & rng (f | X) c= Y ) ;
hence z in { F where F is Function of X,Y : F is one-to-one } ; ::_thesis: verum
end;
then { f where f is Function of X,Y : ( f is one-to-one & rng (f | X) c= Y ) } = { F where F is Function of X,Y : F is one-to-one } by A22, XBOOLE_0:def_10;
hence card { F where F is Function of X,Y : F is one-to-one } = card { F where F is Function of (X \/ {x}),(Y \/ {y}) : ( F is one-to-one & F . x = y ) } by A9, A21, A25, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem :: CARD_FIN:6
for n, k being Nat holds (n !) / ((n -' k) !) is Element of NAT
proof
let n, k be Nat; ::_thesis: (n !) / ((n -' k) !) is Element of NAT
( n in NAT & k in NAT ) by ORDINAL1:def_12;
then (n !) / ((n -' k) !) is integer by IRRAT_1:36, NAT_D:35;
hence (n !) / ((n -' k) !) is Element of NAT by INT_1:3; ::_thesis: verum
end;
theorem Th7: :: CARD_FIN:7
for X, Y being finite set st card X <= card Y holds
card { F where F is Function of X,Y : F is one-to-one } = ((card Y) !) / (((card Y) -' (card X)) !)
proof
let X, Y be finite set ; ::_thesis: ( card X <= card Y implies card { F where F is Function of X,Y : F is one-to-one } = ((card Y) !) / (((card Y) -' (card X)) !) )
defpred S1[ Nat] means for X, Y being finite set st card Y = $1 & card X <= card Y holds
card { F where F is Function of X,Y : F is one-to-one } = ((card Y) !) / (((card Y) -' (card X)) !);
A1: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A2: S1[n] ; ::_thesis: S1[n + 1]
let X, Y be finite set ; ::_thesis: ( card Y = n + 1 & card X <= card Y implies card { F where F is Function of X,Y : F is one-to-one } = ((card Y) !) / (((card Y) -' (card X)) !) )
assume that
A3: card Y = n + 1 and
A4: card X <= card Y ; ::_thesis: card { F where F is Function of X,Y : F is one-to-one } = ((card Y) !) / (((card Y) -' (card X)) !)
percases ( X is empty or not X is empty ) ;
supposeA5: X is empty ; ::_thesis: card { F where F is Function of X,Y : F is one-to-one } = ((card Y) !) / (((card Y) -' (card X)) !)
set F1 = { F where F is Function of X,Y : F is one-to-one } ;
A6: { F where F is Function of X,Y : F is one-to-one } c= {{}}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { F where F is Function of X,Y : F is one-to-one } or x in {{}} )
assume x in { F where F is Function of X,Y : F is one-to-one } ; ::_thesis: x in {{}}
then ex F being Function of X,Y st
( x = F & F is one-to-one ) ;
then x = {} by A5;
hence x in {{}} by TARSKI:def_1; ::_thesis: verum
end;
A7: rng {} c= Y by XBOOLE_1:2;
(card Y) - (card X) = card Y by A5, CARD_1:27;
then A8: ((card Y) -' (card X)) ! = (card Y) ! by XREAL_0:def_2;
(card Y) ! > 0 by NEWTON:17;
then A9: ((card Y) !) / (((card Y) -' (card X)) !) = 1 by A8, XCMPLX_1:60;
dom {} = X by A5;
then {} is Function of X,Y by A7, FUNCT_2:2;
then {} in { F where F is Function of X,Y : F is one-to-one } ;
then { F where F is Function of X,Y : F is one-to-one } = {{}} by A6, ZFMISC_1:33;
hence card { F where F is Function of X,Y : F is one-to-one } = ((card Y) !) / (((card Y) -' (card X)) !) by A9, CARD_1:30; ::_thesis: verum
end;
suppose not X is empty ; ::_thesis: card { F where F is Function of X,Y : F is one-to-one } = ((card Y) !) / (((card Y) -' (card X)) !)
then consider x being set such that
A10: x in X by XBOOLE_0:def_1;
defpred S2[ Function] means $1 is one-to-one ;
card Y,Y are_equipotent by CARD_1:def_2;
then consider f being Function such that
A11: f is one-to-one and
A12: dom f = card Y and
A13: rng f = Y by WELLORD2:def_4;
reconsider f = f as Function of (card Y),Y by A12, A13, FUNCT_2:1;
A14: not Y is empty by A3;
A15: ( f is onto & f is one-to-one ) by A11, A13, FUNCT_2:def_3;
consider F being XFinSequence of such that
A16: dom F = card Y and
A17: card { g where g is Function of X,Y : S2[g] } = Sum F and
A18: for k being Nat st k in dom F holds
F . k = card { g where g is Function of X,Y : ( S2[g] & g . x = f . k ) } from STIRL2_1:sch_6(A15, A14, A10);
A19: for k being Nat st k in dom F holds
F . k = (n !) / (((card Y) -' (card X)) !)
proof
card X > 0 by A10;
then reconsider cX1 = (card X) - 1 as Element of NAT by NAT_1:20;
set Xx = X \ {x};
x in {x} by TARSKI:def_1;
then A20: not x in X \ {x} by XBOOLE_0:def_5;
A21: X = (X \ {x}) \/ {x} by A10, ZFMISC_1:116;
A22: (cX1 + 1) - 1 <= (n + 1) - 1 by A3, A4, XREAL_1:9;
then A23: n - cX1 >= cX1 - cX1 by XREAL_1:9;
let k be Nat; ::_thesis: ( k in dom F implies F . k = (n !) / (((card Y) -' (card X)) !) )
assume A24: k in dom F ; ::_thesis: F . k = (n !) / (((card Y) -' (card X)) !)
A25: f . k in Y by A12, A13, A16, A24, FUNCT_1:def_3;
set Yy = Y \ {(f . k)};
A26: Y = (Y \ {(f . k)}) \/ {(f . k)} by A25, ZFMISC_1:116;
f . k in {(f . k)} by TARSKI:def_1;
then A27: not f . k in Y \ {(f . k)} by XBOOLE_0:def_5;
cX1 + 1 <= n + 1 by A3, A4;
then A28: card (X \ {x}) = cX1 by A10, STIRL2_1:55;
A29: card (Y \ {(f . k)}) = n by A3, A25, STIRL2_1:55;
then A30: ( Y \ {(f . k)} is empty implies X \ {x} is empty ) by A22, A28, CARD_1:27;
A31: card { g where g is Function of (X \ {x}),(Y \ {(f . k)}) : g is one-to-one } = (n !) / (((card (Y \ {(f . k)})) -' (card (X \ {x}))) !) by A2, A22, A28, A29;
(card Y) - (card X) >= (card X) - (card X) by A4, XREAL_1:9;
then (card Y) -' (card X) = (((card (Y \ {(f . k)})) + 1) - 1) - (((card (X \ {x})) + 1) - 1) by A3, A28, A29, XREAL_0:def_2
.= (card (Y \ {(f . k)})) -' (card (X \ {x})) by A28, A29, A23, XREAL_0:def_2 ;
then card { g where g is Function of X,Y : ( g is one-to-one & g . x = f . k ) } = (n !) / (((card Y) -' (card X)) !) by A31, A26, A21, A27, A20, A30, Th5;
hence F . k = (n !) / (((card Y) -' (card X)) !) by A18, A24; ::_thesis: verum
end;
then for k being Nat st k in dom F holds
F . k >= (n !) / (((card Y) -' (card X)) !) ;
then A32: Sum F >= (len F) * ((n !) / (((card Y) -' (card X)) !)) by AFINSQ_2:60;
for k being Nat st k in dom F holds
F . k <= (n !) / (((card Y) -' (card X)) !) by A19;
then Sum F <= (len F) * ((n !) / (((card Y) -' (card X)) !)) by AFINSQ_2:59;
then Sum F = (n + 1) * ((n !) / (((card Y) -' (card X)) !)) by A3, A16, A32, XXREAL_0:1
.= ((n + 1) * (n !)) / (((card Y) -' (card X)) !) by XCMPLX_1:74
.= ((card Y) !) / (((card Y) -' (card X)) !) by A3, NEWTON:15 ;
hence card { F where F is Function of X,Y : F is one-to-one } = ((card Y) !) / (((card Y) -' (card X)) !) by A17; ::_thesis: verum
end;
end;
end;
A33: S1[ 0 ]
proof
let X, Y be finite set ; ::_thesis: ( card Y = 0 & card X <= card Y implies card { F where F is Function of X,Y : F is one-to-one } = ((card Y) !) / (((card Y) -' (card X)) !) )
assume that
A34: card Y = 0 and
A35: card X <= card Y ; ::_thesis: card { F where F is Function of X,Y : F is one-to-one } = ((card Y) !) / (((card Y) -' (card X)) !)
(card Y) - (card X) = 0 by A34, A35;
then A36: ((card Y) -' (card X)) ! = 1 by NEWTON:12, XREAL_0:def_2;
set F1 = { F where F is Function of X,Y : F is one-to-one } ;
A37: { F where F is Function of X,Y : F is one-to-one } c= {{}}
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { F where F is Function of X,Y : F is one-to-one } or x in {{}} )
assume x in { F where F is Function of X,Y : F is one-to-one } ; ::_thesis: x in {{}}
then A38: ex F being Function of X,Y st
( x = F & F is one-to-one ) ;
Y = {} by A34;
then x = {} by A38;
hence x in {{}} by TARSKI:def_1; ::_thesis: verum
end;
( dom {} = X & rng {} = Y ) by A34, A35;
then {} is Function of X,Y by FUNCT_2:1;
then {} in { F where F is Function of X,Y : F is one-to-one } ;
then { F where F is Function of X,Y : F is one-to-one } = {{}} by A37, ZFMISC_1:33;
hence card { F where F is Function of X,Y : F is one-to-one } = ((card Y) !) / (((card Y) -' (card X)) !) by A34, A36, CARD_1:30, NEWTON:12; ::_thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch_2(A33, A1);
hence ( card X <= card Y implies card { F where F is Function of X,Y : F is one-to-one } = ((card Y) !) / (((card Y) -' (card X)) !) ) ; ::_thesis: verum
end;
theorem Th8: :: CARD_FIN:8
for X being finite set holds card { F where F is Function of X,X : F is Permutation of X } = (card X) !
proof
let X be finite set ; ::_thesis: card { F where F is Function of X,X : F is Permutation of X } = (card X) !
set F1 = { F where F is Function of X,X : F is one-to-one } ;
set F2 = { F where F is Function of X,X : F is Permutation of X } ;
A1: { F where F is Function of X,X : F is one-to-one } c= { F where F is Function of X,X : F is Permutation of X }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { F where F is Function of X,X : F is one-to-one } or x in { F where F is Function of X,X : F is Permutation of X } )
assume x in { F where F is Function of X,X : F is one-to-one } ; ::_thesis: x in { F where F is Function of X,X : F is Permutation of X }
then consider F being Function of X,X such that
A2: x = F and
A3: F is one-to-one ;
card X = card X ;
then F is onto by A3, STIRL2_1:60;
hence x in { F where F is Function of X,X : F is Permutation of X } by A2, A3; ::_thesis: verum
end;
((card X) -' (card X)) ! = 1 by NEWTON:12, XREAL_1:232;
then A4: ((card X) !) / (((card X) -' (card X)) !) = (card X) ! ;
{ F where F is Function of X,X : F is Permutation of X } c= { F where F is Function of X,X : F is one-to-one }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { F where F is Function of X,X : F is Permutation of X } or x in { F where F is Function of X,X : F is one-to-one } )
assume x in { F where F is Function of X,X : F is Permutation of X } ; ::_thesis: x in { F where F is Function of X,X : F is one-to-one }
then ex F being Function of X,X st
( x = F & F is Permutation of X ) ;
hence x in { F where F is Function of X,X : F is one-to-one } ; ::_thesis: verum
end;
then { F where F is Function of X,X : F is one-to-one } = { F where F is Function of X,X : F is Permutation of X } by A1, XBOOLE_0:def_10;
hence card { F where F is Function of X,X : F is Permutation of X } = (card X) ! by A4, Th7; ::_thesis: verum
end;
definition
let X be finite set ;
let k be Nat;
let x1, x2 be set ;
func Choose (X,k,x1,x2) -> Subset of (Funcs (X,{x1,x2})) means :Def1: :: CARD_FIN:def 1
for x being set holds
( x in it iff ex f being Function of X,{x1,x2} st
( f = x & card (f " {x1}) = k ) );
existence
ex b1 being Subset of (Funcs (X,{x1,x2})) st
for x being set holds
( x in b1 iff ex f being Function of X,{x1,x2} st
( f = x & card (f " {x1}) = k ) )
proof
defpred S1[ set ] means ex f being Function of X,{x1,x2} st
( $1 = f & card (f " {x1}) = k );
consider F being set such that
A1: for x being set holds
( x in F iff ( x in bool [:X,{x1,x2}:] & S1[x] ) ) from XBOOLE_0:sch_1();
F c= Funcs (X,{x1,x2})
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in F or x in Funcs (X,{x1,x2}) )
assume x in F ; ::_thesis: x in Funcs (X,{x1,x2})
then ex f being Function of X,{x1,x2} st
( x = f & card (f " {x1}) = k ) by A1;
hence x in Funcs (X,{x1,x2}) by FUNCT_2:8; ::_thesis: verum
end;
then reconsider F = F as Subset of (Funcs (X,{x1,x2})) ;
take F ; ::_thesis: for x being set holds
( x in F iff ex f being Function of X,{x1,x2} st
( f = x & card (f " {x1}) = k ) )
let x be set ; ::_thesis: ( x in F iff ex f being Function of X,{x1,x2} st
( f = x & card (f " {x1}) = k ) )
thus ( x in F implies ex f being Function of X,{x1,x2} st
( x = f & card (f " {x1}) = k ) ) by A1; ::_thesis: ( ex f being Function of X,{x1,x2} st
( f = x & card (f " {x1}) = k ) implies x in F )
given f being Function of X,{x1,x2} such that A2: ( x = f & card (f " {x1}) = k ) ; ::_thesis: x in F
thus x in F by A1, A2; ::_thesis: verum
end;
uniqueness
for b1, b2 being Subset of (Funcs (X,{x1,x2})) st ( for x being set holds
( x in b1 iff ex f being Function of X,{x1,x2} st
( f = x & card (f " {x1}) = k ) ) ) & ( for x being set holds
( x in b2 iff ex f being Function of X,{x1,x2} st
( f = x & card (f " {x1}) = k ) ) ) holds
b1 = b2
proof
let F1, F2 be Subset of (Funcs (X,{x1,x2})); ::_thesis: ( ( for x being set holds
( x in F1 iff ex f being Function of X,{x1,x2} st
( f = x & card (f " {x1}) = k ) ) ) & ( for x being set holds
( x in F2 iff ex f being Function of X,{x1,x2} st
( f = x & card (f " {x1}) = k ) ) ) implies F1 = F2 )
assume that
A3: for x being set holds
( x in F1 iff ex f being Function of X,{x1,x2} st
( x = f & card (f " {x1}) = k ) ) and
A4: for x being set holds
( x in F2 iff ex f being Function of X,{x1,x2} st
( x = f & card (f " {x1}) = k ) ) ; ::_thesis: F1 = F2
for x being set holds
( x in F1 iff x in F2 )
proof
let x be set ; ::_thesis: ( x in F1 iff x in F2 )
( x in F1 iff ex f being Function of X,{x1,x2} st
( x = f & card (f " {x1}) = k ) ) by A3;
hence ( x in F1 iff x in F2 ) by A4; ::_thesis: verum
end;
hence F1 = F2 by TARSKI:1; ::_thesis: verum
end;
end;
:: deftheorem Def1 defines Choose CARD_FIN:def_1_:_
for X being finite set
for k being Nat
for x1, x2 being set
for b5 being Subset of (Funcs (X,{x1,x2})) holds
( b5 = Choose (X,k,x1,x2) iff for x being set holds
( x in b5 iff ex f being Function of X,{x1,x2} st
( f = x & card (f " {x1}) = k ) ) );
theorem :: CARD_FIN:9
for x1 being set
for X being finite set
for k being Nat st card X <> k holds
Choose (X,k,x1,x1) is empty
proof
let x1 be set ; ::_thesis: for X being finite set
for k being Nat st card X <> k holds
Choose (X,k,x1,x1) is empty
let X be finite set ; ::_thesis: for k being Nat st card X <> k holds
Choose (X,k,x1,x1) is empty
let k be Nat; ::_thesis: ( card X <> k implies Choose (X,k,x1,x1) is empty )
assume A1: card X <> k ; ::_thesis: Choose (X,k,x1,x1) is empty
assume not Choose (X,k,x1,x1) is empty ; ::_thesis: contradiction
then consider y being set such that
A2: y in Choose (X,k,x1,x1) by XBOOLE_0:def_1;
consider f being Function of X,{x1,x1} such that
f = y and
A3: card (f " {x1}) = k by A2, Def1;
percases ( rng f is empty or not rng f is empty ) ;
supposeA4: rng f is empty ; ::_thesis: contradiction
A5: dom f = X by FUNCT_2:def_1;
dom f = {} by A4, RELAT_1:42;
hence contradiction by A1, A3, A5; ::_thesis: verum
end;
supposeA6: not rng f is empty ; ::_thesis: contradiction
{x1,x1} = {x1} by ENUMSET1:29;
then rng f = {x1} by A6, ZFMISC_1:33;
then f " {x1} = dom f by RELAT_1:134;
hence contradiction by A1, A3, FUNCT_2:def_1; ::_thesis: verum
end;
end;
end;
theorem Th10: :: CARD_FIN:10
for x1, x2 being set
for X being finite set
for k being Nat st card X < k holds
Choose (X,k,x1,x2) is empty
proof
let x1, x2 be set ; ::_thesis: for X being finite set
for k being Nat st card X < k holds
Choose (X,k,x1,x2) is empty
let X be finite set ; ::_thesis: for k being Nat st card X < k holds
Choose (X,k,x1,x2) is empty
let k be Nat; ::_thesis: ( card X < k implies Choose (X,k,x1,x2) is empty )
assume A1: card X < k ; ::_thesis: Choose (X,k,x1,x2) is empty
assume not Choose (X,k,x1,x2) is empty ; ::_thesis: contradiction
then consider z being set such that
A2: z in Choose (X,k,x1,x2) by XBOOLE_0:def_1;
ex f being Function of X,{x1,x2} st
( f = z & card (f " {x1}) = k ) by A2, Def1;
hence contradiction by A1, NAT_1:43; ::_thesis: verum
end;
theorem Th11: :: CARD_FIN:11
for x1, x2 being set
for X being finite set st x1 <> x2 holds
card (Choose (X,0,x1,x2)) = 1
proof
let x1, x2 be set ; ::_thesis: for X being finite set st x1 <> x2 holds
card (Choose (X,0,x1,x2)) = 1
let X be finite set ; ::_thesis: ( x1 <> x2 implies card (Choose (X,0,x1,x2)) = 1 )
assume A1: x1 <> x2 ; ::_thesis: card (Choose (X,0,x1,x2)) = 1
percases ( X is empty or not X is empty ) ;
supposeA2: X is empty ; ::_thesis: card (Choose (X,0,x1,x2)) = 1
dom {} = X by A2;
then reconsider Empty = {} as Function of X,{x1,x2} by XBOOLE_1:2;
A3: Choose (X,0,x1,x2) c= {Empty}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Choose (X,0,x1,x2) or z in {Empty} )
assume z in Choose (X,0,x1,x2) ; ::_thesis: z in {Empty}
then consider f being Function of X,{x1,x2} such that
A4: z = f and
card (f " {x1}) = 0 by Def1;
dom f = X by FUNCT_2:def_1;
then f = Empty ;
hence z in {Empty} by A4, TARSKI:def_1; ::_thesis: verum
end;
( Empty " {x1} = {} & card {} = 0 ) ;
then Empty in Choose (X,0,x1,x2) by Def1;
then Choose (X,0,x1,x2) = {Empty} by A3, ZFMISC_1:33;
hence card (Choose (X,0,x1,x2)) = 1 by CARD_1:30; ::_thesis: verum
end;
supposeA5: not X is empty ; ::_thesis: card (Choose (X,0,x1,x2)) = 1
then consider f being Function such that
A6: dom f = X and
A7: rng f = {x2} by FUNCT_1:5;
rng f c= {x1,x2} by A7, ZFMISC_1:36;
then A8: f is Function of X,{x1,x2} by A6, FUNCT_2:2;
A9: Choose (X,0,x1,x2) c= {f}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Choose (X,0,x1,x2) or z in {f} )
assume z in Choose (X,0,x1,x2) ; ::_thesis: z in {f}
then consider g being Function of X,{x1,x2} such that
A10: z = g and
A11: card (g " {x1}) = 0 by Def1;
g " {x1} = {} by A11;
then not x1 in rng g by FUNCT_1:72;
then ( not rng g = {x1} & not rng g = {x1,x2} ) by TARSKI:def_1, TARSKI:def_2;
then ( dom g = X & rng g = {x2} ) by A5, FUNCT_2:def_1, ZFMISC_1:36;
then g = f by A6, A7, FUNCT_1:7;
hence z in {f} by A10, TARSKI:def_1; ::_thesis: verum
end;
not x1 in rng f by A1, A7, TARSKI:def_1;
then A12: f " {x1} = {} by FUNCT_1:72;
card {} = 0 ;
then f in Choose (X,0,x1,x2) by A12, A8, Def1;
then Choose (X,0,x1,x2) = {f} by A9, ZFMISC_1:33;
hence card (Choose (X,0,x1,x2)) = 1 by CARD_1:30; ::_thesis: verum
end;
end;
end;
theorem Th12: :: CARD_FIN:12
for x1, x2 being set
for X being finite set holds card (Choose (X,(card X),x1,x2)) = 1
proof
let x1, x2 be set ; ::_thesis: for X being finite set holds card (Choose (X,(card X),x1,x2)) = 1
let X be finite set ; ::_thesis: card (Choose (X,(card X),x1,x2)) = 1
percases ( X is empty or not X is empty ) ;
supposeA1: X is empty ; ::_thesis: card (Choose (X,(card X),x1,x2)) = 1
dom {} = X by A1;
then reconsider Empty = {} as Function of X,{x1,x2} by XBOOLE_1:2;
A2: Choose (X,(card X),x1,x2) c= {Empty}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Choose (X,(card X),x1,x2) or z in {Empty} )
assume z in Choose (X,(card X),x1,x2) ; ::_thesis: z in {Empty}
then consider f being Function of X,{x1,x2} such that
A3: z = f and
card (f " {x1}) = card X by Def1;
dom f = X by FUNCT_2:def_1;
then f = Empty ;
hence z in {Empty} by A3, TARSKI:def_1; ::_thesis: verum
end;
Empty " {x1} = {} ;
then Empty in Choose (X,(card X),x1,x2) by A1, Def1;
then Choose (X,(card X),x1,x2) = {Empty} by A2, ZFMISC_1:33;
hence card (Choose (X,(card X),x1,x2)) = 1 by CARD_1:30; ::_thesis: verum
end;
supposeA4: not X is empty ; ::_thesis: card (Choose (X,(card X),x1,x2)) = 1
then consider f being Function such that
A5: dom f = X and
A6: rng f = {x1} by FUNCT_1:5;
rng f c= {x1,x2} by A6, ZFMISC_1:36;
then A7: f is Function of X,{x1,x2} by A5, FUNCT_2:2;
A8: Choose (X,(card X),x1,x2) c= {f}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Choose (X,(card X),x1,x2) or z in {f} )
assume z in Choose (X,(card X),x1,x2) ; ::_thesis: z in {f}
then consider g being Function of X,{x1,x2} such that
A9: z = g and
A10: card (g " {x1}) = card X by Def1;
A11: now__::_thesis:_rng_g_=_{x1}
percases ( x1 = x2 or x1 <> x2 ) ;
suppose x1 = x2 ; ::_thesis: rng g = {x1}
then {x1,x2} = {x1} by ENUMSET1:29;
hence rng g = {x1} by A4, ZFMISC_1:33; ::_thesis: verum
end;
supposeA12: x1 <> x2 ; ::_thesis: rng g = {x1}
g " {x2} = {}
proof
assume g " {x2} <> {} ; ::_thesis: contradiction
then consider z being set such that
A13: z in g " {x2} by XBOOLE_0:def_1;
g . z in {x2} by A13, FUNCT_1:def_7;
then A14: g . z = x2 by TARSKI:def_1;
g " {x1} = X by A10, Th1;
then g . z in {x1} by A13, FUNCT_1:def_7;
hence contradiction by A12, A14, TARSKI:def_1; ::_thesis: verum
end;
then not x2 in rng g by FUNCT_1:72;
then ( not rng g = {x2} & not rng g = {x1,x2} ) by TARSKI:def_1, TARSKI:def_2;
hence rng g = {x1} by A4, ZFMISC_1:36; ::_thesis: verum
end;
end;
end;
dom g = X by FUNCT_2:def_1;
then g = f by A5, A6, A11, FUNCT_1:7;
hence z in {f} by A9, TARSKI:def_1; ::_thesis: verum
end;
card (f " {x1}) = card X by A5, A6, RELAT_1:134;
then f in Choose (X,(card X),x1,x2) by A7, Def1;
then Choose (X,(card X),x1,x2) = {f} by A8, ZFMISC_1:33;
hence card (Choose (X,(card X),x1,x2)) = 1 by CARD_1:30; ::_thesis: verum
end;
end;
end;
theorem Th13: :: CARD_FIN:13
for z, x, y being set
for X being finite set
for k being Nat st not z in X holds
card (Choose (X,k,x,y)) = card { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) }
proof
let z, x, y be set ; ::_thesis: for X being finite set
for k being Nat st not z in X holds
card (Choose (X,k,x,y)) = card { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) }
let X be finite set ; ::_thesis: for k being Nat st not z in X holds
card (Choose (X,k,x,y)) = card { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) }
let k be Nat; ::_thesis: ( not z in X implies card (Choose (X,k,x,y)) = card { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } )
set F1 = { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } ;
defpred S1[ set , set , set ] means for f being Function st f = $1 holds
card ((f | X) " {x}) = k;
A1: for f being Function of (X \/ {z}),({x,y} \/ {x}) st f . z = x holds
( S1[f,X \/ {z},{x,y} \/ {x}] iff S1[f | X,X,{x,y}] )
proof
let f be Function of (X \/ {z}),({x,y} \/ {x}); ::_thesis: ( f . z = x implies ( S1[f,X \/ {z},{x,y} \/ {x}] iff S1[f | X,X,{x,y}] ) )
assume f . z = x ; ::_thesis: ( S1[f,X \/ {z},{x,y} \/ {x}] iff S1[f | X,X,{x,y}] )
f | X = (f | X) | X ;
hence ( S1[f,X \/ {z},{x,y} \/ {x}] iff S1[f | X,X,{x,y}] ) ; ::_thesis: verum
end;
set F3 = { f where f is Function of (X \/ {z}),({x,y} \/ {x}) : ( S1[f,X \/ {z},{x,y} \/ {x}] & rng (f | X) c= {x,y} & f . z = x ) } ;
set F2 = { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } ;
assume A2: not z in X ; ::_thesis: card (Choose (X,k,x,y)) = card { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) }
A3: { f where f is Function of (X \/ {z}),({x,y} \/ {x}) : ( S1[f,X \/ {z},{x,y} \/ {x}] & rng (f | X) c= {x,y} & f . z = x ) } c= { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) }
proof
{x} \/ {x,y} = {x,x,y} by ENUMSET1:2;
then A4: {x,y} \/ {x} = {x,y} by ENUMSET1:30;
z in {z} by TARSKI:def_1;
then A5: z in X \/ {z} by XBOOLE_0:def_3;
let x1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in { f where f is Function of (X \/ {z}),({x,y} \/ {x}) : ( S1[f,X \/ {z},{x,y} \/ {x}] & rng (f | X) c= {x,y} & f . z = x ) } or x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } )
assume x1 in { f where f is Function of (X \/ {z}),({x,y} \/ {x}) : ( S1[f,X \/ {z},{x,y} \/ {x}] & rng (f | X) c= {x,y} & f . z = x ) } ; ::_thesis: x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) }
then consider f being Function of (X \/ {z}),({x,y} \/ {x}) such that
A6: x1 = f and
A7: S1[f,X \/ {z},{x,y} \/ {x}] and
rng (f | X) c= {x,y} and
A8: f . z = x ;
( dom f = X \/ {z} & (X \/ {z}) \ {z} = X ) by A2, FUNCT_2:def_1, ZFMISC_1:117;
then A9: {z} \/ ((f | X) " {x}) = f " {x} by A8, A5, AFINSQ_2:66;
not z in (dom f) /\ X by A2, XBOOLE_0:def_4;
then not z in dom (f | X) by RELAT_1:61;
then A10: not z in (f | X) " {x} by FUNCT_1:def_7;
card ((f | X) " {x}) = k by A7;
then card (f " {x}) = k + 1 by A9, A10, CARD_2:41;
hence x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } by A6, A8, A4; ::_thesis: verum
end;
A11: { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } c= Choose (X,k,x,y)
proof
let x1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } or x1 in Choose (X,k,x,y) )
assume x1 in { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } ; ::_thesis: x1 in Choose (X,k,x,y)
then consider f being Function of X,{x,y} such that
A12: x1 = f and
A13: S1[f,X,{x,y}] ;
f | X = f ;
then card (f " {x}) = k by A13;
hence x1 in Choose (X,k,x,y) by A12, Def1; ::_thesis: verum
end;
A14: Choose (X,k,x,y) c= { f where f is Function of X,{x,y} : S1[f,X,{x,y}] }
proof
let x1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in Choose (X,k,x,y) or x1 in { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } )
assume x1 in Choose (X,k,x,y) ; ::_thesis: x1 in { f where f is Function of X,{x,y} : S1[f,X,{x,y}] }
then consider f being Function of X,{x,y} such that
A15: x1 = f and
A16: card (f " {x}) = k by Def1;
dom f = X by FUNCT_2:def_1;
then S1[f,X,{x,y}] by A16, RELAT_1:68;
hence x1 in { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } by A15; ::_thesis: verum
end;
A17: ( {x,y} is empty implies X is empty ) ;
A18: card { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } = card { f where f is Function of (X \/ {z}),({x,y} \/ {x}) : ( S1[f,X \/ {z},{x,y} \/ {x}] & rng (f | X) c= {x,y} & f . z = x ) } from STIRL2_1:sch_4(A17, A2, A1);
{ f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } c= { f where f is Function of (X \/ {z}),({x,y} \/ {x}) : ( S1[f,X \/ {z},{x,y} \/ {x}] & rng (f | X) c= {x,y} & f . z = x ) }
proof
z in {z} by TARSKI:def_1;
then A19: z in X \/ {z} by XBOOLE_0:def_3;
let x1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } or x1 in { f where f is Function of (X \/ {z}),({x,y} \/ {x}) : ( S1[f,X \/ {z},{x,y} \/ {x}] & rng (f | X) c= {x,y} & f . z = x ) } )
assume x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } ; ::_thesis: x1 in { f where f is Function of (X \/ {z}),({x,y} \/ {x}) : ( S1[f,X \/ {z},{x,y} \/ {x}] & rng (f | X) c= {x,y} & f . z = x ) }
then consider f being Function of (X \/ {z}),{x,y} such that
A20: x1 = f and
A21: card (f " {x}) = k + 1 and
A22: f . z = x ;
not z in (dom f) /\ X by A2, XBOOLE_0:def_4;
then not z in dom (f | X) by RELAT_1:61;
then A23: not z in (f | X) " {x} by FUNCT_1:def_7;
( dom f = X \/ {z} & (X \/ {z}) \ {z} = X ) by A2, FUNCT_2:def_1, ZFMISC_1:117;
then ((f | X) " {x}) \/ {z} = f " {x} by A22, A19, AFINSQ_2:66;
then (card ((f | X) " {x})) + 1 = k + 1 by A21, A23, CARD_2:41;
then A24: S1[f,X \/ {z},{x,y} \/ {x}] ;
{x} \/ {x,y} = {x,x,y} by ENUMSET1:2;
then ( rng (f | X) c= {x,y} & f is Function of (X \/ {z}),({x,y} \/ {x}) ) by ENUMSET1:30;
hence x1 in { f where f is Function of (X \/ {z}),({x,y} \/ {x}) : ( S1[f,X \/ {z},{x,y} \/ {x}] & rng (f | X) c= {x,y} & f . z = x ) } by A20, A22, A24; ::_thesis: verum
end;
then { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } = { f where f is Function of (X \/ {z}),({x,y} \/ {x}) : ( S1[f,X \/ {z},{x,y} \/ {x}] & rng (f | X) c= {x,y} & f . z = x ) } by A3, XBOOLE_0:def_10;
hence card (Choose (X,k,x,y)) = card { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } by A11, A14, A18, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th14: :: CARD_FIN:14
for z, x, y being set
for X being finite set
for k being Nat st not z in X & x <> y holds
card (Choose (X,k,x,y)) = card { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) }
proof
let z, x, y be set ; ::_thesis: for X being finite set
for k being Nat st not z in X & x <> y holds
card (Choose (X,k,x,y)) = card { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) }
let X be finite set ; ::_thesis: for k being Nat st not z in X & x <> y holds
card (Choose (X,k,x,y)) = card { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) }
let k be Nat; ::_thesis: ( not z in X & x <> y implies card (Choose (X,k,x,y)) = card { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) } )
assume that
A1: not z in X and
A2: x <> y ; ::_thesis: card (Choose (X,k,x,y)) = card { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) }
defpred S1[ set , set , set ] means for f being Function st f = $1 holds
card (f " {x}) = k;
A3: for f being Function of (X \/ {z}),({x,y} \/ {y}) st f . z = y holds
( S1[f,X \/ {z},{x,y} \/ {y}] iff S1[f | X,X,{x,y}] )
proof
let f be Function of (X \/ {z}),({x,y} \/ {y}); ::_thesis: ( f . z = y implies ( S1[f,X \/ {z},{x,y} \/ {y}] iff S1[f | X,X,{x,y}] ) )
assume A4: f . z = y ; ::_thesis: ( S1[f,X \/ {z},{x,y} \/ {y}] iff S1[f | X,X,{x,y}] )
( (X \/ {z}) \ {z} = X & dom f = X \/ {z} ) by A1, FUNCT_2:def_1, ZFMISC_1:117;
then (f | X) " {x} = f " {x} by A2, A4, AFINSQ_2:67;
hence ( S1[f,X \/ {z},{x,y} \/ {y}] iff S1[f | X,X,{x,y}] ) ; ::_thesis: verum
end;
set F2 = { f where f is Function of (X \/ {z}),({x,y} \/ {y}) : ( S1[f,X \/ {z},{x,y} \/ {y}] & rng (f | X) c= {x,y} & f . z = y ) } ;
set F1 = { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } ;
A5: ( {x,y} is empty implies X is empty ) ;
A6: card { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } = card { f where f is Function of (X \/ {z}),({x,y} \/ {y}) : ( S1[f,X \/ {z},{x,y} \/ {y}] & rng (f | X) c= {x,y} & f . z = y ) } from STIRL2_1:sch_4(A5, A1, A3);
set F3 = { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) } ;
A7: { f where f is Function of (X \/ {z}),({x,y} \/ {y}) : ( S1[f,X \/ {z},{x,y} \/ {y}] & rng (f | X) c= {x,y} & f . z = y ) } c= { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) }
proof
let x1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in { f where f is Function of (X \/ {z}),({x,y} \/ {y}) : ( S1[f,X \/ {z},{x,y} \/ {y}] & rng (f | X) c= {x,y} & f . z = y ) } or x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) } )
assume x1 in { f where f is Function of (X \/ {z}),({x,y} \/ {y}) : ( S1[f,X \/ {z},{x,y} \/ {y}] & rng (f | X) c= {x,y} & f . z = y ) } ; ::_thesis: x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) }
then consider f being Function of (X \/ {z}),({x,y} \/ {y}) such that
A8: x1 = f and
A9: S1[f,X \/ {z},{x,y} \/ {y}] and
rng (f | X) c= {x,y} and
A10: f . z = y ;
{x,y} \/ {y} = {y,y,x} by ENUMSET1:2;
then A11: f is Function of (X \/ {z}),{x,y} by ENUMSET1:30;
card (f " {x}) = k by A9;
hence x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) } by A8, A10, A11; ::_thesis: verum
end;
A12: { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) } c= { f where f is Function of (X \/ {z}),({x,y} \/ {y}) : ( S1[f,X \/ {z},{x,y} \/ {y}] & rng (f | X) c= {x,y} & f . z = y ) }
proof
let x1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) } or x1 in { f where f is Function of (X \/ {z}),({x,y} \/ {y}) : ( S1[f,X \/ {z},{x,y} \/ {y}] & rng (f | X) c= {x,y} & f . z = y ) } )
assume x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) } ; ::_thesis: x1 in { f where f is Function of (X \/ {z}),({x,y} \/ {y}) : ( S1[f,X \/ {z},{x,y} \/ {y}] & rng (f | X) c= {x,y} & f . z = y ) }
then consider f being Function of (X \/ {z}),{x,y} such that
A13: f = x1 and
A14: card (f " {x}) = k and
A15: f . z = y ;
{x,y} \/ {y} = {y,y,x} by ENUMSET1:2;
then A16: ( rng (f | X) c= {x,y} & f is Function of (X \/ {z}),({x,y} \/ {y}) ) by ENUMSET1:30;
S1[f,X \/ {z},{x,y} \/ {y}] by A14;
hence x1 in { f where f is Function of (X \/ {z}),({x,y} \/ {y}) : ( S1[f,X \/ {z},{x,y} \/ {y}] & rng (f | X) c= {x,y} & f . z = y ) } by A13, A15, A16; ::_thesis: verum
end;
A17: Choose (X,k,x,y) c= { f where f is Function of X,{x,y} : S1[f,X,{x,y}] }
proof
let x1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in Choose (X,k,x,y) or x1 in { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } )
assume x1 in Choose (X,k,x,y) ; ::_thesis: x1 in { f where f is Function of X,{x,y} : S1[f,X,{x,y}] }
then consider f being Function of X,{x,y} such that
A18: x1 = f and
A19: card (f " {x}) = k by Def1;
S1[f,X,{x,y}] by A19;
hence x1 in { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } by A18; ::_thesis: verum
end;
{ f where f is Function of X,{x,y} : S1[f,X,{x,y}] } c= Choose (X,k,x,y)
proof
let x1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } or x1 in Choose (X,k,x,y) )
assume x1 in { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } ; ::_thesis: x1 in Choose (X,k,x,y)
then consider f being Function of X,{x,y} such that
A20: x1 = f and
A21: S1[f,X,{x,y}] ;
card (f " {x}) = k by A21;
hence x1 in Choose (X,k,x,y) by A20, Def1; ::_thesis: verum
end;
then Choose (X,k,x,y) = { f where f is Function of X,{x,y} : S1[f,X,{x,y}] } by A17, XBOOLE_0:def_10;
hence card (Choose (X,k,x,y)) = card { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k & f . z = y ) } by A7, A12, A6, XBOOLE_0:def_10; ::_thesis: verum
end;
Lm1: for x, y, z being set
for X being finite set
for k being Nat st x <> y holds
( { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } = Choose ((X \/ {z}),(k + 1),x,y) & { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } misses { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } )
proof
let x, y, z be set ; ::_thesis: for X being finite set
for k being Nat st x <> y holds
( { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } = Choose ((X \/ {z}),(k + 1),x,y) & { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } misses { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } )
let X be finite set ; ::_thesis: for k being Nat st x <> y holds
( { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } = Choose ((X \/ {z}),(k + 1),x,y) & { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } misses { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } )
let k be Nat; ::_thesis: ( x <> y implies ( { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } = Choose ((X \/ {z}),(k + 1),x,y) & { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } misses { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } ) )
set F1 = { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } ;
set F2 = { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } ;
assume A1: x <> y ; ::_thesis: ( { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } = Choose ((X \/ {z}),(k + 1),x,y) & { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } misses { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } )
A2: { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } misses { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) }
proof
assume { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } meets { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } ; ::_thesis: contradiction
then { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } /\ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } <> {} by XBOOLE_0:def_7;
then consider x1 being set such that
A3: x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } /\ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } by XBOOLE_0:def_1;
x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } by A3, XBOOLE_0:def_4;
then A4: ex f2 being Function of (X \/ {z}),{x,y} st
( x1 = f2 & card (f2 " {x}) = k + 1 & f2 . z = y ) ;
x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } by A3, XBOOLE_0:def_4;
then ex f1 being Function of (X \/ {z}),{x,y} st
( x1 = f1 & card (f1 " {x}) = k + 1 & f1 . z = x ) ;
hence contradiction by A1, A4; ::_thesis: verum
end;
A5: { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } c= Choose ((X \/ {z}),(k + 1),x,y)
proof
let x1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } or x1 in Choose ((X \/ {z}),(k + 1),x,y) )
assume x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } ; ::_thesis: x1 in Choose ((X \/ {z}),(k + 1),x,y)
then ex f being Function of (X \/ {z}),{x,y} st
( x1 = f & card (f " {x}) = k + 1 & f . z = y ) ;
hence x1 in Choose ((X \/ {z}),(k + 1),x,y) by Def1; ::_thesis: verum
end;
A6: Choose ((X \/ {z}),(k + 1),x,y) c= { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) }
proof
let x1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in Choose ((X \/ {z}),(k + 1),x,y) or x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } )
assume x1 in Choose ((X \/ {z}),(k + 1),x,y) ; ::_thesis: x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) }
then consider f being Function of (X \/ {z}),{x,y} such that
A7: ( f = x1 & card (f " {x}) = k + 1 ) by Def1;
z in {z} by TARSKI:def_1;
then A8: z in X \/ {z} by XBOOLE_0:def_3;
dom f = X \/ {z} by FUNCT_2:def_1;
then f . z in rng f by A8, FUNCT_1:def_3;
then ( f . z = x or f . z = y ) by TARSKI:def_2;
then ( x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } or x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } ) by A7;
hence x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } by XBOOLE_0:def_3; ::_thesis: verum
end;
{ f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } c= Choose ((X \/ {z}),(k + 1),x,y)
proof
let x1 be set ; :: according to TARSKI:def_3 ::_thesis: ( not x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } or x1 in Choose ((X \/ {z}),(k + 1),x,y) )
assume x1 in { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } ; ::_thesis: x1 in Choose ((X \/ {z}),(k + 1),x,y)
then ex f being Function of (X \/ {z}),{x,y} st
( x1 = f & card (f " {x}) = k + 1 & f . z = x ) ;
hence x1 in Choose ((X \/ {z}),(k + 1),x,y) by Def1; ::_thesis: verum
end;
then { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } c= Choose ((X \/ {z}),(k + 1),x,y) by A5, XBOOLE_1:8;
hence ( { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } = Choose ((X \/ {z}),(k + 1),x,y) & { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } misses { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } ) by A6, A2, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th15: :: CARD_FIN:15
for x, y, z being set
for X being finite set
for k being Nat st x <> y & not z in X holds
card (Choose ((X \/ {z}),(k + 1),x,y)) = (card (Choose (X,(k + 1),x,y))) + (card (Choose (X,k,x,y)))
proof
let x, y, z be set ; ::_thesis: for X being finite set
for k being Nat st x <> y & not z in X holds
card (Choose ((X \/ {z}),(k + 1),x,y)) = (card (Choose (X,(k + 1),x,y))) + (card (Choose (X,k,x,y)))
let X be finite set ; ::_thesis: for k being Nat st x <> y & not z in X holds
card (Choose ((X \/ {z}),(k + 1),x,y)) = (card (Choose (X,(k + 1),x,y))) + (card (Choose (X,k,x,y)))
let k be Nat; ::_thesis: ( x <> y & not z in X implies card (Choose ((X \/ {z}),(k + 1),x,y)) = (card (Choose (X,(k + 1),x,y))) + (card (Choose (X,k,x,y))) )
assume that
A1: x <> y and
A2: not z in X ; ::_thesis: card (Choose ((X \/ {z}),(k + 1),x,y)) = (card (Choose (X,(k + 1),x,y))) + (card (Choose (X,k,x,y)))
set F2 = { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } ;
set F1 = { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } ;
A3: { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } = Choose ((X \/ {z}),(k + 1),x,y) by A1, Lm1;
( { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } c= { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } & { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } c= { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } \/ { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } ) by XBOOLE_1:7;
then reconsider F1 = { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = x ) } , F2 = { f where f is Function of (X \/ {z}),{x,y} : ( card (f " {x}) = k + 1 & f . z = y ) } as finite set by A3;
A4: card F1 = card (Choose (X,k,x,y)) by A2, Th13;
( card (F1 \/ F2) = (card F1) + (card F2) & card F2 = card (Choose (X,(k + 1),x,y)) ) by A1, A2, Lm1, Th14, CARD_2:40;
hence card (Choose ((X \/ {z}),(k + 1),x,y)) = (card (Choose (X,(k + 1),x,y))) + (card (Choose (X,k,x,y))) by A1, A4, Lm1; ::_thesis: verum
end;
theorem Th16: :: CARD_FIN:16
for x, y being set
for X being finite set
for k being Nat st x <> y holds
card (Choose (X,k,x,y)) = (card X) choose k
proof
let x, y be set ; ::_thesis: for X being finite set
for k being Nat st x <> y holds
card (Choose (X,k,x,y)) = (card X) choose k
let X be finite set ; ::_thesis: for k being Nat st x <> y holds
card (Choose (X,k,x,y)) = (card X) choose k
let k be Nat; ::_thesis: ( x <> y implies card (Choose (X,k,x,y)) = (card X) choose k )
defpred S1[ Nat] means for k being Nat
for X being finite set st k + (card X) <= $1 holds
card (Choose (X,k,x,y)) = (card X) choose k;
assume A1: x <> y ; ::_thesis: card (Choose (X,k,x,y)) = (card X) choose k
A2: for n being Nat st S1[n] holds
S1[n + 1]
proof
let n be Nat; ::_thesis: ( S1[n] implies S1[n + 1] )
assume A3: S1[n] ; ::_thesis: S1[n + 1]
let k be Nat; ::_thesis: for X being finite set st k + (card X) <= n + 1 holds
card (Choose (X,k,x,y)) = (card X) choose k
let X be finite set ; ::_thesis: ( k + (card X) <= n + 1 implies card (Choose (X,k,x,y)) = (card X) choose k )
assume A4: k + (card X) <= n + 1 ; ::_thesis: card (Choose (X,k,x,y)) = (card X) choose k
percases ( k + (card X) < n + 1 or k + (card X) = n + 1 ) by A4, XXREAL_0:1;
suppose k + (card X) < n + 1 ; ::_thesis: card (Choose (X,k,x,y)) = (card X) choose k
then k + (card X) <= n by NAT_1:13;
hence card (Choose (X,k,x,y)) = (card X) choose k by A3; ::_thesis: verum
end;
supposeA5: k + (card X) = n + 1 ; ::_thesis: card (Choose (X,k,x,y)) = (card X) choose k
percases ( ( k = 0 & card X >= 0 ) or ( k > 0 & card X = 0 ) or ( k > 0 & card X > 0 ) ) ;
supposeA6: ( k = 0 & card X >= 0 ) ; ::_thesis: card (Choose (X,k,x,y)) = (card X) choose k
then card (Choose (X,k,x,y)) = 1 by A1, Th11;
hence card (Choose (X,k,x,y)) = (card X) choose k by A6, NEWTON:19; ::_thesis: verum
end;
supposeA7: ( k > 0 & card X = 0 ) ; ::_thesis: card (Choose (X,k,x,y)) = (card X) choose k
then Choose (X,k,x,y) is empty by Th10;
hence card (Choose (X,k,x,y)) = (card X) choose k by A7, NEWTON:def_3; ::_thesis: verum
end;
supposeA8: ( k > 0 & card X > 0 ) ; ::_thesis: card (Choose (X,k,x,y)) = (card X) choose k
then reconsider cXz = (card X) - 1 as Element of NAT by NAT_1:20;
reconsider k1 = k - 1 as Element of NAT by A8, NAT_1:20;
consider z being set such that
A9: z in X by A8, CARD_1:27, XBOOLE_0:def_1;
set Xz = X \ {z};
z in {z} by TARSKI:def_1;
then A10: not z in X \ {z} by XBOOLE_0:def_5;
(X \ {z}) \/ {z} = X by A9, ZFMISC_1:116;
then A11: card (Choose (X,(k1 + 1),x,y)) = (card (Choose ((X \ {z}),(k1 + 1),x,y))) + (card (Choose ((X \ {z}),k1,x,y))) by A1, A10, Th15;
card X = cXz + 1 ;
then A12: card (X \ {z}) = cXz by A9, STIRL2_1:55;
cXz < cXz + 1 by NAT_1:13;
then A13: card (X \ {z}) < card X by A9, STIRL2_1:55;
then k + (card (X \ {z})) < n + 1 by A5, XREAL_1:8;
then k + (card (X \ {z})) <= n by NAT_1:13;
then A14: card (Choose ((X \ {z}),(k1 + 1),x,y)) = (card (X \ {z})) choose (k1 + 1) by A3;
k1 < k1 + 1 by NAT_1:13;
then k1 + (card (X \ {z})) < n + 1 by A5, A13, XREAL_1:8;
then k1 + (card (X \ {z})) <= n by NAT_1:13;
then A15: card (Choose ((X \ {z}),k1,x,y)) = (card (X \ {z})) choose k1 by A3;
card X = cXz + 1 ;
hence card (Choose (X,k,x,y)) = (card X) choose k by A14, A15, A11, A12, NEWTON:22; ::_thesis: verum
end;
end;
end;
end;
end;
A16: S1[ 0 ]
proof
let k be Nat; ::_thesis: for X being finite set st k + (card X) <= 0 holds
card (Choose (X,k,x,y)) = (card X) choose k
let X be finite set ; ::_thesis: ( k + (card X) <= 0 implies card (Choose (X,k,x,y)) = (card X) choose k )
A17: 0 choose 0 = 1 by NEWTON:19;
assume k + (card X) <= 0 ; ::_thesis: card (Choose (X,k,x,y)) = (card X) choose k
then ( k + (card X) = 0 & card X = 0 ) ;
hence card (Choose (X,k,x,y)) = (card X) choose k by A1, Th11, A17; ::_thesis: verum
end;
for n being Nat holds S1[n] from NAT_1:sch_2(A16, A2);
then S1[k + (card X)] ;
hence card (Choose (X,k,x,y)) = (card X) choose k ; ::_thesis: verum
end;
theorem Th17: :: CARD_FIN:17
for x, y being set
for Y, X being finite set st x <> y holds
(Y --> y) +* (X --> x) in Choose ((X \/ Y),(card X),x,y)
proof
let x, y be set ; ::_thesis: for Y, X being finite set st x <> y holds
(Y --> y) +* (X --> x) in Choose ((X \/ Y),(card X),x,y)
let Y, X be finite set ; ::_thesis: ( x <> y implies (Y --> y) +* (X --> x) in Choose ((X \/ Y),(card X),x,y) )
set F = (Y --> y) +* (X --> x);
( dom (Y --> y) = Y & dom (X --> x) = X ) by FUNCOP_1:13;
then A1: dom ((Y --> y) +* (X --> x)) = X \/ Y by FUNCT_4:def_1;
{y} c= {x,y} by ZFMISC_1:7;
then A2: rng (Y --> y) c= {x,y} by XBOOLE_1:1;
{x} c= {x,y} by ZFMISC_1:7;
then rng (X --> x) c= {x,y} by XBOOLE_1:1;
then ( rng ((Y --> y) +* (X --> x)) c= (rng (X --> x)) \/ (rng (Y --> y)) & (rng (X --> x)) \/ (rng (Y --> y)) c= {x,y} ) by A2, FUNCT_4:17, XBOOLE_1:8;
then reconsider F = (Y --> y) +* (X --> x) as Function of (X \/ Y),{x,y} by A1, FUNCT_2:2, XBOOLE_1:1;
assume A3: x <> y ; ::_thesis: (Y --> y) +* (X --> x) in Choose ((X \/ Y),(card X),x,y)
A4: F " {x} c= X
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in F " {x} or z in X )
assume A5: z in F " {x} ; ::_thesis: z in X
A6: ( z in X or z in Y ) by A5, XBOOLE_0:def_3;
F . z in {x} by A5, FUNCT_1:def_7;
then A7: F . z = x by TARSKI:def_1;
z in dom F by A5, FUNCT_1:def_7;
then A8: z in (dom (X --> x)) \/ (dom (Y --> y)) by FUNCT_4:def_1;
assume A9: not z in X ; ::_thesis: contradiction
X = dom (X --> x) by FUNCOP_1:13;
then F . z = (Y --> y) . z by A9, A8, FUNCT_4:def_1;
hence contradiction by A3, A9, A6, A7, FUNCOP_1:7; ::_thesis: verum
end;
X c= F " {x}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in X or z in F " {x} )
assume A10: z in X ; ::_thesis: z in F " {x}
A11: z in dom F by A1, A10, XBOOLE_0:def_3;
z in dom (X --> x) by A10, FUNCOP_1:13;
then A12: F . z = (X --> x) . z by FUNCT_4:13;
(X --> x) . z = x by A10, FUNCOP_1:7;
then F . z in {x} by A12, TARSKI:def_1;
hence z in F " {x} by A11, FUNCT_1:def_7; ::_thesis: verum
end;
then X = F " {x} by A4, XBOOLE_0:def_10;
hence (Y --> y) +* (X --> x) in Choose ((X \/ Y),(card X),x,y) by Def1; ::_thesis: verum
end;
theorem Th18: :: CARD_FIN:18
for x, y being set
for X, Y being finite set st x <> y & X misses Y holds
(X --> x) +* (Y --> y) in Choose ((X \/ Y),(card X),x,y)
proof
let x, y be set ; ::_thesis: for X, Y being finite set st x <> y & X misses Y holds
(X --> x) +* (Y --> y) in Choose ((X \/ Y),(card X),x,y)
let X, Y be finite set ; ::_thesis: ( x <> y & X misses Y implies (X --> x) +* (Y --> y) in Choose ((X \/ Y),(card X),x,y) )
assume that
A1: x <> y and
A2: X misses Y ; ::_thesis: (X --> x) +* (Y --> y) in Choose ((X \/ Y),(card X),x,y)
( dom (X --> x) = X & dom (Y --> y) = Y ) by FUNCOP_1:13;
then (X --> x) +* (Y --> y) = (Y --> y) +* (X --> x) by A2, FUNCT_4:35;
hence (X --> x) +* (Y --> y) in Choose ((X \/ Y),(card X),x,y) by A1, Th17; ::_thesis: verum
end;
definition
let F, Ch be Function;
let y be set ;
func Intersection (F,Ch,y) -> Subset of (union (rng F)) means :Def2: :: CARD_FIN:def 2
for z being set holds
( z in it iff ( z in union (rng F) & ( for x being set st x in dom Ch & Ch . x = y holds
z in F . x ) ) );
existence
ex b1 being Subset of (union (rng F)) st
for z being set holds
( z in b1 iff ( z in union (rng F) & ( for x being set st x in dom Ch & Ch . x = y holds
z in F . x ) ) )
proof
defpred S1[ set ] means for x being set st x in dom Ch & Ch . x = y holds
$1 in F . x;
consider I being set such that
A1: for z being set holds
( z in I iff ( z in union (rng F) & S1[z] ) ) from XBOOLE_0:sch_1();
I c= union (rng F)
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in I or x in union (rng F) )
assume x in I ; ::_thesis: x in union (rng F)
hence x in union (rng F) by A1; ::_thesis: verum
end;
then reconsider I = I as Subset of (union (rng F)) ;
take I ; ::_thesis: for z being set holds
( z in I iff ( z in union (rng F) & ( for x being set st x in dom Ch & Ch . x = y holds
z in F . x ) ) )
thus for z being set holds
( z in I iff ( z in union (rng F) & ( for x being set st x in dom Ch & Ch . x = y holds
z in F . x ) ) ) by A1; ::_thesis: verum
end;
uniqueness
for b1, b2 being Subset of (union (rng F)) st ( for z being set holds
( z in b1 iff ( z in union (rng F) & ( for x being set st x in dom Ch & Ch . x = y holds
z in F . x ) ) ) ) & ( for z being set holds
( z in b2 iff ( z in union (rng F) & ( for x being set st x in dom Ch & Ch . x = y holds
z in F . x ) ) ) ) holds
b1 = b2
proof
let I1, I2 be Subset of (union (rng F)); ::_thesis: ( ( for z being set holds
( z in I1 iff ( z in union (rng F) & ( for x being set st x in dom Ch & Ch . x = y holds
z in F . x ) ) ) ) & ( for z being set holds
( z in I2 iff ( z in union (rng F) & ( for x being set st x in dom Ch & Ch . x = y holds
z in F . x ) ) ) ) implies I1 = I2 )
assume that
A2: for z being set holds
( z in I1 iff ( z in union (rng F) & ( for x being set st x in dom Ch & Ch . x = y holds
z in F . x ) ) ) and
A3: for z being set holds
( z in I2 iff ( z in union (rng F) & ( for x being set st x in dom Ch & Ch . x = y holds
z in F . x ) ) ) ; ::_thesis: I1 = I2
for z being set holds
( z in I1 iff z in I2 )
proof
let z be set ; ::_thesis: ( z in I1 iff z in I2 )
( z in I1 iff ( z in union (rng F) & ( for x being set st x in dom Ch & Ch . x = y holds
z in F . x ) ) ) by A2;
hence ( z in I1 iff z in I2 ) by A3; ::_thesis: verum
end;
hence I1 = I2 by TARSKI:1; ::_thesis: verum
end;
end;
:: deftheorem Def2 defines Intersection CARD_FIN:def_2_:_
for F, Ch being Function
for y being set
for b4 being Subset of (union (rng F)) holds
( b4 = Intersection (F,Ch,y) iff for z being set holds
( z in b4 iff ( z in union (rng F) & ( for x being set st x in dom Ch & Ch . x = y holds
z in F . x ) ) ) );
theorem Th19: :: CARD_FIN:19
for x, y being set
for F, Ch being Function st not (dom F) /\ (Ch " {x}) is empty holds
( y in Intersection (F,Ch,x) iff for z being set st z in dom Ch & Ch . z = x holds
y in F . z )
proof
let x, y be set ; ::_thesis: for F, Ch being Function st not (dom F) /\ (Ch " {x}) is empty holds
( y in Intersection (F,Ch,x) iff for z being set st z in dom Ch & Ch . z = x holds
y in F . z )
let F, Ch be Function; ::_thesis: ( not (dom F) /\ (Ch " {x}) is empty implies ( y in Intersection (F,Ch,x) iff for z being set st z in dom Ch & Ch . z = x holds
y in F . z ) )
assume A1: not (dom F) /\ (Ch " {x}) is empty ; ::_thesis: ( y in Intersection (F,Ch,x) iff for z being set st z in dom Ch & Ch . z = x holds
y in F . z )
thus ( y in Intersection (F,Ch,x) implies for z being set st z in dom Ch & Ch . z = x holds
y in F . z ) by Def2; ::_thesis: ( ( for z being set st z in dom Ch & Ch . z = x holds
y in F . z ) implies y in Intersection (F,Ch,x) )
thus ( ( for z being set st z in dom Ch & Ch . z = x holds
y in F . z ) implies y in Intersection (F,Ch,x) ) ::_thesis: verum
proof
consider z being set such that
A2: z in (dom F) /\ (Ch " {x}) by A1, XBOOLE_0:def_1;
A3: z in Ch " {x} by A2, XBOOLE_0:def_4;
then Ch . z in {x} by FUNCT_1:def_7;
then A4: Ch . z = x by TARSKI:def_1;
z in dom F by A2, XBOOLE_0:def_4;
then A5: F . z in rng F by FUNCT_1:def_3;
assume A6: for z being set st z in dom Ch & Ch . z = x holds
y in F . z ; ::_thesis: y in Intersection (F,Ch,x)
z in dom Ch by A3, FUNCT_1:def_7;
then y in F . z by A6, A4;
then y in union (rng F) by A5, TARSKI:def_4;
hence y in Intersection (F,Ch,x) by A6, Def2; ::_thesis: verum
end;
end;
theorem Th20: :: CARD_FIN:20
for y being set
for F, Ch being Function st not Intersection (F,Ch,y) is empty holds
Ch " {y} c= dom F
proof
let y be set ; ::_thesis: for F, Ch being Function st not Intersection (F,Ch,y) is empty holds
Ch " {y} c= dom F
let F, Ch be Function; ::_thesis: ( not Intersection (F,Ch,y) is empty implies Ch " {y} c= dom F )
assume not Intersection (F,Ch,y) is empty ; ::_thesis: Ch " {y} c= dom F
then consider z being set such that
A1: z in Intersection (F,Ch,y) by XBOOLE_0:def_1;
assume not Ch " {y} c= dom F ; ::_thesis: contradiction
then consider x being set such that
A2: x in Ch " {y} and
A3: not x in dom F by TARSKI:def_3;
Ch . x in {y} by A2, FUNCT_1:def_7;
then A4: Ch . x = y by TARSKI:def_1;
x in dom Ch by A2, FUNCT_1:def_7;
then z in F . x by A1, A4, Def2;
hence contradiction by A3, FUNCT_1:def_2; ::_thesis: verum
end;
theorem :: CARD_FIN:21
for y being set
for F, Ch being Function st not Intersection (F,Ch,y) is empty holds
for x1, x2 being set st x1 in Ch " {y} & x2 in Ch " {y} holds
F . x1 meets F . x2
proof
let y be set ; ::_thesis: for F, Ch being Function st not Intersection (F,Ch,y) is empty holds
for x1, x2 being set st x1 in Ch " {y} & x2 in Ch " {y} holds
F . x1 meets F . x2
let F, Ch be Function; ::_thesis: ( not Intersection (F,Ch,y) is empty implies for x1, x2 being set st x1 in Ch " {y} & x2 in Ch " {y} holds
F . x1 meets F . x2 )
assume not Intersection (F,Ch,y) is empty ; ::_thesis: for x1, x2 being set st x1 in Ch " {y} & x2 in Ch " {y} holds
F . x1 meets F . x2
then consider z being set such that
A1: z in Intersection (F,Ch,y) by XBOOLE_0:def_1;
let x1, x2 be set ; ::_thesis: ( x1 in Ch " {y} & x2 in Ch " {y} implies F . x1 meets F . x2 )
assume that
A2: x1 in Ch " {y} and
A3: x2 in Ch " {y} ; ::_thesis: F . x1 meets F . x2
Ch . x2 in {y} by A3, FUNCT_1:def_7;
then A4: Ch . x2 = y by TARSKI:def_1;
Ch . x1 in {y} by A2, FUNCT_1:def_7;
then A5: Ch . x1 = y by TARSKI:def_1;
x2 in dom Ch by A3, FUNCT_1:def_7;
then A6: z in F . x2 by A1, A4, Def2;
x1 in dom Ch by A2, FUNCT_1:def_7;
then z in F . x1 by A1, A5, Def2;
hence F . x1 meets F . x2 by A6, XBOOLE_0:3; ::_thesis: verum
end;
theorem Th22: :: CARD_FIN:22
for z, y being set
for F, Ch being Function st z in Intersection (F,Ch,y) & y in rng Ch holds
ex x being set st
( x in dom Ch & Ch . x = y & z in F . x )
proof
let z, y be set ; ::_thesis: for F, Ch being Function st z in Intersection (F,Ch,y) & y in rng Ch holds
ex x being set st
( x in dom Ch & Ch . x = y & z in F . x )
let F, Ch be Function; ::_thesis: ( z in Intersection (F,Ch,y) & y in rng Ch implies ex x being set st
( x in dom Ch & Ch . x = y & z in F . x ) )
assume that
A1: z in Intersection (F,Ch,y) and
A2: y in rng Ch ; ::_thesis: ex x being set st
( x in dom Ch & Ch . x = y & z in F . x )
Ch " {y} <> {} by A2, FUNCT_1:72;
then consider x being set such that
A3: x in Ch " {y} by XBOOLE_0:def_1;
Ch . x in {y} by A3, FUNCT_1:def_7;
then A4: Ch . x = y by TARSKI:def_1;
A5: x in dom Ch by A3, FUNCT_1:def_7;
x in dom Ch by A3, FUNCT_1:def_7;
then z in F . x by A1, A4, Def2;
hence ex x being set st
( x in dom Ch & Ch . x = y & z in F . x ) by A4, A5; ::_thesis: verum
end;
theorem :: CARD_FIN:23
for y being set
for F, Ch being Function st ( F is empty or union (rng F) is empty ) holds
Intersection (F,Ch,y) = union (rng F) by RELAT_1:38, ZFMISC_1:2;
theorem Th24: :: CARD_FIN:24
for y being set
for F, Ch being Function st F | (Ch " {y}) = (Ch " {y}) --> (union (rng F)) holds
Intersection (F,Ch,y) = union (rng F)
proof
let y be set ; ::_thesis: for F, Ch being Function st F | (Ch " {y}) = (Ch " {y}) --> (union (rng F)) holds
Intersection (F,Ch,y) = union (rng F)
let F, Ch be Function; ::_thesis: ( F | (Ch " {y}) = (Ch " {y}) --> (union (rng F)) implies Intersection (F,Ch,y) = union (rng F) )
set ChF = (Ch " {y}) --> (union (rng F));
assume A1: F | (Ch " {y}) = (Ch " {y}) --> (union (rng F)) ; ::_thesis: Intersection (F,Ch,y) = union (rng F)
union (rng F) c= Intersection (F,Ch,y)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in union (rng F) or z in Intersection (F,Ch,y) )
assume A2: z in union (rng F) ; ::_thesis: z in Intersection (F,Ch,y)
now__::_thesis:_for_x_being_set_st_x_in_dom_Ch_&_Ch_._x_=_y_holds_
z_in_F_._x
let x be set ; ::_thesis: ( x in dom Ch & Ch . x = y implies z in F . x )
assume that
A3: x in dom Ch and
A4: Ch . x = y ; ::_thesis: z in F . x
Ch . x in {y} by A4, TARSKI:def_1;
then A5: x in Ch " {y} by A3, FUNCT_1:def_7;
then ( dom ((Ch " {y}) --> (union (rng F))) = Ch " {y} & ((Ch " {y}) --> (union (rng F))) . x = union (rng F) ) by FUNCOP_1:7, FUNCOP_1:13;
hence z in F . x by A1, A2, A5, FUNCT_1:47; ::_thesis: verum
end;
hence z in Intersection (F,Ch,y) by A2, Def2; ::_thesis: verum
end;
hence Intersection (F,Ch,y) = union (rng F) by XBOOLE_0:def_10; ::_thesis: verum
end;
theorem :: CARD_FIN:25
for y being set
for F, Ch being Function st not union (rng F) is empty & Intersection (F,Ch,y) = union (rng F) holds
F | (Ch " {y}) = (Ch " {y}) --> (union (rng F))
proof
let y be set ; ::_thesis: for F, Ch being Function st not union (rng F) is empty & Intersection (F,Ch,y) = union (rng F) holds
F | (Ch " {y}) = (Ch " {y}) --> (union (rng F))
let F, Ch be Function; ::_thesis: ( not union (rng F) is empty & Intersection (F,Ch,y) = union (rng F) implies F | (Ch " {y}) = (Ch " {y}) --> (union (rng F)) )
set ChF = (Ch " {y}) --> (union (rng F));
assume that
A1: not union (rng F) is empty and
A2: Intersection (F,Ch,y) = union (rng F) ; ::_thesis: F | (Ch " {y}) = (Ch " {y}) --> (union (rng F))
A3: (dom F) /\ (Ch " {y}) = dom (F | (Ch " {y})) by RELAT_1:61;
(dom F) /\ (Ch " {y}) = Ch " {y} by A1, A2, Th20, XBOOLE_1:28;
then A4: dom (F | (Ch " {y})) = dom ((Ch " {y}) --> (union (rng F))) by A3, FUNCOP_1:13;
assume F | (Ch " {y}) <> (Ch " {y}) --> (union (rng F)) ; ::_thesis: contradiction
then consider x being set such that
A5: x in dom (F | (Ch " {y})) and
A6: (F | (Ch " {y})) . x <> ((Ch " {y}) --> (union (rng F))) . x by A4, FUNCT_1:2;
x in (dom F) /\ (Ch " {y}) by A5, RELAT_1:61;
then A7: x in dom F by XBOOLE_0:def_4;
x in (dom F) /\ (Ch " {y}) by A5, RELAT_1:61;
then A8: x in Ch " {y} by XBOOLE_0:def_4;
then A9: ((Ch " {y}) --> (union (rng F))) . x = union (rng F) by FUNCOP_1:7;
Ch . x in {y} by A8, FUNCT_1:def_7;
then A10: Ch . x = y by TARSKI:def_1;
F . x = (F | (Ch " {y})) . x by A5, FUNCT_1:47;
then (F | (Ch " {y})) . x in rng F by A7, FUNCT_1:def_3;
then (F | (Ch " {y})) . x c= ((Ch " {y}) --> (union (rng F))) . x by A9, ZFMISC_1:74;
then not union (rng F) c= (F | (Ch " {y})) . x by A6, A9, XBOOLE_0:def_10;
then consider z being set such that
A11: z in union (rng F) and
A12: not z in (F | (Ch " {y})) . x by TARSKI:def_3;
x in dom Ch by A8, FUNCT_1:def_7;
then z in F . x by A2, A11, A10, Def2;
hence contradiction by A5, A12, FUNCT_1:47; ::_thesis: verum
end;
theorem Th26: :: CARD_FIN:26
for y being set
for F being Function holds Intersection (F,{},y) = union (rng F)
proof
let y be set ; ::_thesis: for F being Function holds Intersection (F,{},y) = union (rng F)
let F be Function; ::_thesis: Intersection (F,{},y) = union (rng F)
F | ({} " {y}) = ({} " {y}) --> (union (rng F)) ;
hence Intersection (F,{},y) = union (rng F) by Th24; ::_thesis: verum
end;
theorem Th27: :: CARD_FIN:27
for y, X9 being set
for F, Ch being Function holds Intersection (F,Ch,y) c= Intersection (F,(Ch | X9),y)
proof
let y, X9 be set ; ::_thesis: for F, Ch being Function holds Intersection (F,Ch,y) c= Intersection (F,(Ch | X9),y)
let F, Ch be Function; ::_thesis: Intersection (F,Ch,y) c= Intersection (F,(Ch | X9),y)
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Intersection (F,Ch,y) or z in Intersection (F,(Ch | X9),y) )
assume A1: z in Intersection (F,Ch,y) ; ::_thesis: z in Intersection (F,(Ch | X9),y)
now__::_thesis:_for_x_being_set_st_x_in_dom_(Ch_|_X9)_&_(Ch_|_X9)_._x_=_y_holds_
z_in_F_._x
let x be set ; ::_thesis: ( x in dom (Ch | X9) & (Ch | X9) . x = y implies z in F . x )
assume that
A2: x in dom (Ch | X9) and
A3: (Ch | X9) . x = y ; ::_thesis: z in F . x
x in (dom Ch) /\ X9 by A2, RELAT_1:61;
then A4: x in dom Ch by XBOOLE_0:def_4;
Ch . x = y by A2, A3, FUNCT_1:47;
hence z in F . x by A1, A4, Def2; ::_thesis: verum
end;
hence z in Intersection (F,(Ch | X9),y) by A1, Def2; ::_thesis: verum
end;
theorem Th28: :: CARD_FIN:28
for y, X9 being set
for Ch, F being Function st Ch " {y} = (Ch | X9) " {y} holds
Intersection (F,Ch,y) = Intersection (F,(Ch | X9),y)
proof
let y, X9 be set ; ::_thesis: for Ch, F being Function st Ch " {y} = (Ch | X9) " {y} holds
Intersection (F,Ch,y) = Intersection (F,(Ch | X9),y)
let Ch, F be Function; ::_thesis: ( Ch " {y} = (Ch | X9) " {y} implies Intersection (F,Ch,y) = Intersection (F,(Ch | X9),y) )
assume A1: Ch " {y} = (Ch | X9) " {y} ; ::_thesis: Intersection (F,Ch,y) = Intersection (F,(Ch | X9),y)
A2: Intersection (F,(Ch | X9),y) c= Intersection (F,Ch,y)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Intersection (F,(Ch | X9),y) or z in Intersection (F,Ch,y) )
assume A3: z in Intersection (F,(Ch | X9),y) ; ::_thesis: z in Intersection (F,Ch,y)
now__::_thesis:_for_x_being_set_st_x_in_dom_Ch_&_Ch_._x_=_y_holds_
z_in_F_._x
let x be set ; ::_thesis: ( x in dom Ch & Ch . x = y implies z in F . x )
assume that
A4: x in dom Ch and
A5: Ch . x = y ; ::_thesis: z in F . x
Ch . x in {y} by A5, TARSKI:def_1;
then A6: x in (Ch | X9) " {y} by A1, A4, FUNCT_1:def_7;
then (Ch | X9) . x in {y} by FUNCT_1:def_7;
then A7: (Ch | X9) . x = y by TARSKI:def_1;
x in dom (Ch | X9) by A6, FUNCT_1:def_7;
hence z in F . x by A3, A7, Def2; ::_thesis: verum
end;
hence z in Intersection (F,Ch,y) by A3, Def2; ::_thesis: verum
end;
Intersection (F,Ch,y) c= Intersection (F,(Ch | X9),y) by Th27;
hence Intersection (F,Ch,y) = Intersection (F,(Ch | X9),y) by A2, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th29: :: CARD_FIN:29
for X9, y being set
for F, Ch being Function holds Intersection ((F | X9),Ch,y) c= Intersection (F,Ch,y)
proof
let X9, y be set ; ::_thesis: for F, Ch being Function holds Intersection ((F | X9),Ch,y) c= Intersection (F,Ch,y)
let F, Ch be Function; ::_thesis: Intersection ((F | X9),Ch,y) c= Intersection (F,Ch,y)
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Intersection ((F | X9),Ch,y) or z in Intersection (F,Ch,y) )
assume A1: z in Intersection ((F | X9),Ch,y) ; ::_thesis: z in Intersection (F,Ch,y)
A2: now__::_thesis:_for_x_being_set_st_x_in_dom_Ch_&_Ch_._x_=_y_holds_
z_in_F_._x
A3: Ch " {y} c= dom (F | X9) by A1, Th20;
let x be set ; ::_thesis: ( x in dom Ch & Ch . x = y implies z in F . x )
assume that
A4: x in dom Ch and
A5: Ch . x = y ; ::_thesis: z in F . x
Ch . x in {y} by A5, TARSKI:def_1;
then A6: x in Ch " {y} by A4, FUNCT_1:def_7;
z in (F | X9) . x by A1, A4, A5, Def2;
hence z in F . x by A6, A3, FUNCT_1:47; ::_thesis: verum
end;
( union (rng (F | X9)) c= union (rng F) & z in union (rng (F | X9)) ) by A1, RELAT_1:70, ZFMISC_1:77;
hence z in Intersection (F,Ch,y) by A2, Def2; ::_thesis: verum
end;
theorem Th30: :: CARD_FIN:30
for y, X9 being set
for Ch, F being Function st y in rng Ch & Ch " {y} c= X9 holds
Intersection ((F | X9),Ch,y) = Intersection (F,Ch,y)
proof
let y, X9 be set ; ::_thesis: for Ch, F being Function st y in rng Ch & Ch " {y} c= X9 holds
Intersection ((F | X9),Ch,y) = Intersection (F,Ch,y)
let Ch, F be Function; ::_thesis: ( y in rng Ch & Ch " {y} c= X9 implies Intersection ((F | X9),Ch,y) = Intersection (F,Ch,y) )
assume that
A1: y in rng Ch and
A2: Ch " {y} c= X9 ; ::_thesis: Intersection ((F | X9),Ch,y) = Intersection (F,Ch,y)
A3: Intersection (F,Ch,y) c= Intersection ((F | X9),Ch,y)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Intersection (F,Ch,y) or z in Intersection ((F | X9),Ch,y) )
assume A4: z in Intersection (F,Ch,y) ; ::_thesis: z in Intersection ((F | X9),Ch,y)
A5: now__::_thesis:_for_x_being_set_st_x_in_dom_Ch_&_Ch_._x_=_y_holds_
z_in_(F_|_X9)_._x
let x be set ; ::_thesis: ( x in dom Ch & Ch . x = y implies z in (F | X9) . x )
assume that
A6: x in dom Ch and
A7: Ch . x = y ; ::_thesis: z in (F | X9) . x
Ch . x in {y} by A7, TARSKI:def_1;
then A8: x in Ch " {y} by A6, FUNCT_1:def_7;
z in F . x by A4, A6, A7, Def2;
then x in dom F by FUNCT_1:def_2;
then x in (dom F) /\ X9 by A2, A8, XBOOLE_0:def_4;
then A9: x in dom (F | X9) by RELAT_1:61;
z in F . x by A4, A6, A7, Def2;
hence z in (F | X9) . x by A9, FUNCT_1:47; ::_thesis: verum
end;
consider x being set such that
A10: x in dom Ch and
A11: Ch . x = y and
A12: z in F . x by A1, A4, Th22;
Ch . x in {y} by A11, TARSKI:def_1;
then A13: x in Ch " {y} by A10, FUNCT_1:def_7;
x in dom F by A12, FUNCT_1:def_2;
then x in (dom F) /\ X9 by A2, A13, XBOOLE_0:def_4;
then x in dom (F | X9) by RELAT_1:61;
then A14: (F | X9) . x in rng (F | X9) by FUNCT_1:def_3;
z in (F | X9) . x by A5, A10, A11;
then z in union (rng (F | X9)) by A14, TARSKI:def_4;
hence z in Intersection ((F | X9),Ch,y) by A5, Def2; ::_thesis: verum
end;
Intersection ((F | X9),Ch,y) c= Intersection (F,Ch,y) by Th29;
hence Intersection ((F | X9),Ch,y) = Intersection (F,Ch,y) by A3, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th31: :: CARD_FIN:31
for x, y being set
for Ch, F being Function st x in Ch " {y} holds
Intersection (F,Ch,y) c= F . x
proof
let x, y be set ; ::_thesis: for Ch, F being Function st x in Ch " {y} holds
Intersection (F,Ch,y) c= F . x
let Ch, F be Function; ::_thesis: ( x in Ch " {y} implies Intersection (F,Ch,y) c= F . x )
assume A1: x in Ch " {y} ; ::_thesis: Intersection (F,Ch,y) c= F . x
then A2: x in dom Ch by FUNCT_1:def_7;
Ch . x in {y} by A1, FUNCT_1:def_7;
then A3: Ch . x = y by TARSKI:def_1;
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Intersection (F,Ch,y) or z in F . x )
assume z in Intersection (F,Ch,y) ; ::_thesis: z in F . x
hence z in F . x by A2, A3, Def2; ::_thesis: verum
end;
theorem Th32: :: CARD_FIN:32
for x, y being set
for Ch, F being Function st x in Ch " {y} holds
(Intersection (F,(Ch | ((dom Ch) \ {x})),y)) /\ (F . x) = Intersection (F,Ch,y)
proof
let x, y be set ; ::_thesis: for Ch, F being Function st x in Ch " {y} holds
(Intersection (F,(Ch | ((dom Ch) \ {x})),y)) /\ (F . x) = Intersection (F,Ch,y)
let Ch, F be Function; ::_thesis: ( x in Ch " {y} implies (Intersection (F,(Ch | ((dom Ch) \ {x})),y)) /\ (F . x) = Intersection (F,Ch,y) )
set d = (dom Ch) \ {x};
set Chd = Ch | ((dom Ch) \ {x});
set I1 = Intersection (F,Ch,y);
set I2 = Intersection (F,(Ch | ((dom Ch) \ {x})),y);
assume x in Ch " {y} ; ::_thesis: (Intersection (F,(Ch | ((dom Ch) \ {x})),y)) /\ (F . x) = Intersection (F,Ch,y)
then A1: Intersection (F,Ch,y) c= F . x by Th31;
A2: (Intersection (F,(Ch | ((dom Ch) \ {x})),y)) /\ (F . x) c= Intersection (F,Ch,y)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in (Intersection (F,(Ch | ((dom Ch) \ {x})),y)) /\ (F . x) or z in Intersection (F,Ch,y) )
assume A3: z in (Intersection (F,(Ch | ((dom Ch) \ {x})),y)) /\ (F . x) ; ::_thesis: z in Intersection (F,Ch,y)
now__::_thesis:_for_x1_being_set_st_x1_in_dom_Ch_&_Ch_._x1_=_y_holds_
z_in_F_._x1
let x1 be set ; ::_thesis: ( x1 in dom Ch & Ch . x1 = y implies z in F . b1 )
assume that
A4: x1 in dom Ch and
A5: Ch . x1 = y ; ::_thesis: z in F . b1
percases ( x1 in (dom Ch) \ {x} or x1 in {x} ) by A4, XBOOLE_0:def_5;
supposeA6: x1 in (dom Ch) \ {x} ; ::_thesis: z in F . b1
A7: z in Intersection (F,(Ch | ((dom Ch) \ {x})),y) by A3, XBOOLE_0:def_4;
A8: ( (dom Ch) /\ ((dom Ch) \ {x}) = dom (Ch | ((dom Ch) \ {x})) & (dom Ch) /\ ((dom Ch) \ {x}) = (dom Ch) \ {x} ) by RELAT_1:61, XBOOLE_1:28;
then (Ch | ((dom Ch) \ {x})) . x1 = y by A5, A6, FUNCT_1:47;
hence z in F . x1 by A6, A8, A7, Def2; ::_thesis: verum
end;
suppose x1 in {x} ; ::_thesis: z in F . b1
then x1 = x by TARSKI:def_1;
hence z in F . x1 by A3, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
end;
hence z in Intersection (F,Ch,y) by A3, Def2; ::_thesis: verum
end;
Intersection (F,Ch,y) c= Intersection (F,(Ch | ((dom Ch) \ {x})),y) by Th27;
then Intersection (F,Ch,y) c= (Intersection (F,(Ch | ((dom Ch) \ {x})),y)) /\ (F . x) by A1, XBOOLE_1:19;
hence (Intersection (F,(Ch | ((dom Ch) \ {x})),y)) /\ (F . x) = Intersection (F,Ch,y) by A2, XBOOLE_0:def_10; ::_thesis: verum
end;
theorem Th33: :: CARD_FIN:33
for x1, x2 being set
for F, Ch1, Ch2 being Function st Ch1 " {x1} = Ch2 " {x2} holds
Intersection (F,Ch1,x1) = Intersection (F,Ch2,x2)
proof
let x1, x2 be set ; ::_thesis: for F, Ch1, Ch2 being Function st Ch1 " {x1} = Ch2 " {x2} holds
Intersection (F,Ch1,x1) = Intersection (F,Ch2,x2)
let F be Function; ::_thesis: for Ch1, Ch2 being Function st Ch1 " {x1} = Ch2 " {x2} holds
Intersection (F,Ch1,x1) = Intersection (F,Ch2,x2)
let Ch1, Ch2 be Function; ::_thesis: ( Ch1 " {x1} = Ch2 " {x2} implies Intersection (F,Ch1,x1) = Intersection (F,Ch2,x2) )
assume A1: Ch1 " {x1} = Ch2 " {x2} ; ::_thesis: Intersection (F,Ch1,x1) = Intersection (F,Ch2,x2)
thus Intersection (F,Ch1,x1) c= Intersection (F,Ch2,x2) :: according to XBOOLE_0:def_10 ::_thesis: Intersection (F,Ch2,x2) c= Intersection (F,Ch1,x1)
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Intersection (F,Ch1,x1) or z in Intersection (F,Ch2,x2) )
assume A2: z in Intersection (F,Ch1,x1) ; ::_thesis: z in Intersection (F,Ch2,x2)
now__::_thesis:_for_x_being_set_st_x_in_dom_Ch2_&_Ch2_._x_=_x2_holds_
z_in_F_._x
let x be set ; ::_thesis: ( x in dom Ch2 & Ch2 . x = x2 implies z in F . x )
assume that
A3: x in dom Ch2 and
A4: Ch2 . x = x2 ; ::_thesis: z in F . x
Ch2 . x in {x2} by A4, TARSKI:def_1;
then A5: x in Ch1 " {x1} by A1, A3, FUNCT_1:def_7;
then Ch1 . x in {x1} by FUNCT_1:def_7;
then A6: Ch1 . x = x1 by TARSKI:def_1;
x in dom Ch1 by A5, FUNCT_1:def_7;
hence z in F . x by A2, A6, Def2; ::_thesis: verum
end;
hence z in Intersection (F,Ch2,x2) by A2, Def2; ::_thesis: verum
end;
thus Intersection (F,Ch2,x2) c= Intersection (F,Ch1,x1) ::_thesis: verum
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Intersection (F,Ch2,x2) or z in Intersection (F,Ch1,x1) )
assume A7: z in Intersection (F,Ch2,x2) ; ::_thesis: z in Intersection (F,Ch1,x1)
now__::_thesis:_for_x_being_set_st_x_in_dom_Ch1_&_Ch1_._x_=_x1_holds_
z_in_F_._x
let x be set ; ::_thesis: ( x in dom Ch1 & Ch1 . x = x1 implies z in F . x )
assume that
A8: x in dom Ch1 and
A9: Ch1 . x = x1 ; ::_thesis: z in F . x
Ch1 . x in {x1} by A9, TARSKI:def_1;
then A10: x in Ch2 " {x2} by A1, A8, FUNCT_1:def_7;
then Ch2 . x in {x2} by FUNCT_1:def_7;
then A11: Ch2 . x = x2 by TARSKI:def_1;
x in dom Ch2 by A10, FUNCT_1:def_7;
hence z in F . x by A7, A11, Def2; ::_thesis: verum
end;
hence z in Intersection (F,Ch1,x1) by A7, Def2; ::_thesis: verum
end;
end;
theorem Th34: :: CARD_FIN:34
for y being set
for Ch, F being Function st Ch " {y} = {} holds
Intersection (F,Ch,y) = union (rng F)
proof
let y be set ; ::_thesis: for Ch, F being Function st Ch " {y} = {} holds
Intersection (F,Ch,y) = union (rng F)
let Ch, F be Function; ::_thesis: ( Ch " {y} = {} implies Intersection (F,Ch,y) = union (rng F) )
reconsider E = {} as set ;
A1: ( Ch | E = {} & Intersection (F,{},y) = union (rng F) ) by Th26;
assume Ch " {y} = {} ; ::_thesis: Intersection (F,Ch,y) = union (rng F)
then (Ch | E) " {y} = Ch " {y} ;
hence Intersection (F,Ch,y) = union (rng F) by A1, Th33; ::_thesis: verum
end;
theorem Th35: :: CARD_FIN:35
for x, y being set
for Ch, F being Function st {x} = Ch " {y} holds
Intersection (F,Ch,y) = F . x
proof
let x, y be set ; ::_thesis: for Ch, F being Function st {x} = Ch " {y} holds
Intersection (F,Ch,y) = F . x
let Ch, F be Function; ::_thesis: ( {x} = Ch " {y} implies Intersection (F,Ch,y) = F . x )
A1: (dom Ch) \ {x} misses {x} by XBOOLE_1:79;
assume A2: {x} = Ch " {y} ; ::_thesis: Intersection (F,Ch,y) = F . x
then (Ch | ((dom Ch) \ {x})) " {y} = ((dom Ch) \ {x}) /\ {x} by FUNCT_1:70;
then (Ch | ((dom Ch) \ {x})) " {y} = {} by A1, XBOOLE_0:def_7;
then A3: Intersection (F,(Ch | ((dom Ch) \ {x})),y) = union (rng F) by Th34;
x in Ch " {y} by A2, TARSKI:def_1;
then A4: (union (rng F)) /\ (F . x) = Intersection (F,Ch,y) by A3, Th32;
percases ( x in dom F or not x in dom F ) ;
suppose x in dom F ; ::_thesis: Intersection (F,Ch,y) = F . x
then F . x in rng F by FUNCT_1:def_3;
hence Intersection (F,Ch,y) = F . x by A4, XBOOLE_1:28, ZFMISC_1:74; ::_thesis: verum
end;
suppose not x in dom F ; ::_thesis: Intersection (F,Ch,y) = F . x
then F . x = {} by FUNCT_1:def_2;
hence Intersection (F,Ch,y) = F . x by A4; ::_thesis: verum
end;
end;
end;
theorem Th36: :: CARD_FIN:36
for x1, x2, y being set
for Ch, F being Function st {x1,x2} = Ch " {y} holds
Intersection (F,Ch,y) = (F . x1) /\ (F . x2)
proof
let x1, x2, y be set ; ::_thesis: for Ch, F being Function st {x1,x2} = Ch " {y} holds
Intersection (F,Ch,y) = (F . x1) /\ (F . x2)
let Ch, F be Function; ::_thesis: ( {x1,x2} = Ch " {y} implies Intersection (F,Ch,y) = (F . x1) /\ (F . x2) )
assume A1: {x1,x2} = Ch " {y} ; ::_thesis: Intersection (F,Ch,y) = (F . x1) /\ (F . x2)
percases ( x1 = x2 or x1 <> x2 ) ;
supposeA2: x1 = x2 ; ::_thesis: Intersection (F,Ch,y) = (F . x1) /\ (F . x2)
then Ch " {y} = {x1} by A1, ENUMSET1:29;
hence Intersection (F,Ch,y) = (F . x1) /\ (F . x2) by A2, Th35; ::_thesis: verum
end;
supposeA3: x1 <> x2 ; ::_thesis: Intersection (F,Ch,y) = (F . x1) /\ (F . x2)
( (Ch " {y}) /\ ((dom Ch) \ {x1}) = ((Ch " {y}) /\ (dom Ch)) \ {x1} & (Ch " {y}) /\ (dom Ch) = {x1,x2} ) by A1, RELAT_1:132, XBOOLE_1:28, XBOOLE_1:49;
then (Ch " {y}) /\ ((dom Ch) \ {x1}) = {x2} by A3, ZFMISC_1:17;
then A4: (Ch | ((dom Ch) \ {x1})) " {y} = {x2} by FUNCT_1:70;
x1 in Ch " {y} by A1, TARSKI:def_2;
then (Intersection (F,(Ch | ((dom Ch) \ {x1})),y)) /\ (F . x1) = Intersection (F,Ch,y) by Th32;
hence Intersection (F,Ch,y) = (F . x1) /\ (F . x2) by A4, Th35; ::_thesis: verum
end;
end;
end;
theorem :: CARD_FIN:37
for y, x being set
for F being Function st not F is empty holds
( y in Intersection (F,((dom F) --> x),x) iff for z being set st z in dom F holds
y in F . z )
proof
let y, x be set ; ::_thesis: for F being Function st not F is empty holds
( y in Intersection (F,((dom F) --> x),x) iff for z being set st z in dom F holds
y in F . z )
let F be Function; ::_thesis: ( not F is empty implies ( y in Intersection (F,((dom F) --> x),x) iff for z being set st z in dom F holds
y in F . z ) )
assume A1: not F is empty ; ::_thesis: ( y in Intersection (F,((dom F) --> x),x) iff for z being set st z in dom F holds
y in F . z )
set Ch = (dom F) --> x;
thus ( y in Intersection (F,((dom F) --> x),x) implies for z being set st z in dom F holds
y in F . z ) ::_thesis: ( ( for z being set st z in dom F holds
y in F . z ) implies y in Intersection (F,((dom F) --> x),x) )
proof
assume A2: y in Intersection (F,((dom F) --> x),x) ; ::_thesis: for z being set st z in dom F holds
y in F . z
let z be set ; ::_thesis: ( z in dom F implies y in F . z )
assume z in dom F ; ::_thesis: y in F . z
then ( z in dom ((dom F) --> x) & ((dom F) --> x) . z = x ) by FUNCOP_1:7, FUNCOP_1:13;
hence y in F . z by A2, Def2; ::_thesis: verum
end;
((dom F) --> x) " {x} = dom F by FUNCOP_1:15;
then A3: (dom F) /\ (((dom F) --> x) " {x}) = dom F ;
assume for z being set st z in dom F holds
y in F . z ; ::_thesis: y in Intersection (F,((dom F) --> x),x)
then for z being set st z in dom ((dom F) --> x) & ((dom F) --> x) . z = x holds
y in F . z ;
hence y in Intersection (F,((dom F) --> x),x) by A1, A3, Th19; ::_thesis: verum
end;
registration
let F be finite-yielding Function;
let X be set ;
clusterF | X -> finite-yielding ;
coherence
F | X is finite-yielding
proof
let x be set ; :: according to FINSET_1:def_4 ::_thesis: ( not x in dom (F | X) or (F | X) . x is finite )
assume x in dom (F | X) ; ::_thesis: (F | X) . x is finite
then (F | X) . x = F . x by FUNCT_1:47;
hence (F | X) . x is finite ; ::_thesis: verum
end;
end;
registration
let F be finite-yielding Function;
let G be Function;
clusterG (#) F -> finite-yielding ;
coherence
F * G is finite-yielding
proof
let x be set ; :: according to FINSET_1:def_4 ::_thesis: ( not x in dom (F * G) or (F * G) . x is finite )
assume x in dom (F * G) ; ::_thesis: (F * G) . x is finite
then (F * G) . x = F . (G . x) by FUNCT_1:12;
hence (F * G) . x is finite ; ::_thesis: verum
end;
cluster Intersect (F,G) -> finite-yielding ;
coherence
Intersect (F,G) is finite-yielding
proof
let x be set ; :: according to FINSET_1:def_4 ::_thesis: ( not x in dom (Intersect (F,G)) or (Intersect (F,G)) . x is finite )
assume x in dom (Intersect (F,G)) ; ::_thesis: (Intersect (F,G)) . x is finite
then x in (dom F) /\ (dom G) by YELLOW20:def_2;
then (Intersect (F,G)) . x = (F . x) /\ (G . x) by YELLOW20:def_2;
hence (Intersect (F,G)) . x is finite ; ::_thesis: verum
end;
end;
theorem :: CARD_FIN:38
for y being set
for Ch being Function
for Fy being finite-yielding Function st y in rng Ch holds
Intersection (Fy,Ch,y) is finite
proof
let y be set ; ::_thesis: for Ch being Function
for Fy being finite-yielding Function st y in rng Ch holds
Intersection (Fy,Ch,y) is finite
let Ch be Function; ::_thesis: for Fy being finite-yielding Function st y in rng Ch holds
Intersection (Fy,Ch,y) is finite
let Fy be finite-yielding Function; ::_thesis: ( y in rng Ch implies Intersection (Fy,Ch,y) is finite )
assume y in rng Ch ; ::_thesis: Intersection (Fy,Ch,y) is finite
then consider x being set such that
A1: x in dom Ch and
A2: Ch . x = y by FUNCT_1:def_3;
Ch . x in {y} by A2, TARSKI:def_1;
then x in Ch " {y} by A1, FUNCT_1:def_7;
then Intersection (Fy,Ch,y) c= Fy . x by Th31;
hence Intersection (Fy,Ch,y) is finite ; ::_thesis: verum
end;
theorem Th39: :: CARD_FIN:39
for Fy being finite-yielding Function st dom Fy is finite holds
union (rng Fy) is finite
proof
let Fy be finite-yielding Function; ::_thesis: ( dom Fy is finite implies union (rng Fy) is finite )
assume dom Fy is finite ; ::_thesis: union (rng Fy) is finite
then rng Fy is finite by FINSET_1:8;
hence union (rng Fy) is finite ; ::_thesis: verum
end;
theorem :: CARD_FIN:40
for x being set
for n, k being Nat holds
( x in Choose (n,k,1,0) iff ex F being XFinSequence of st
( F = x & dom F = n & rng F c= {0,1} & Sum F = k ) )
proof
let x be set ; ::_thesis: for n, k being Nat holds
( x in Choose (n,k,1,0) iff ex F being XFinSequence of st
( F = x & dom F = n & rng F c= {0,1} & Sum F = k ) )
let n, k be Nat; ::_thesis: ( x in Choose (n,k,1,0) iff ex F being XFinSequence of st
( F = x & dom F = n & rng F c= {0,1} & Sum F = k ) )
thus ( x in Choose (n,k,1,0) implies ex F being XFinSequence of st
( F = x & dom F = n & rng F c= {0,1} & Sum F = k ) ) ::_thesis: ( ex F being XFinSequence of st
( F = x & dom F = n & rng F c= {0,1} & Sum F = k ) implies x in Choose (n,k,1,0) )
proof
assume x in Choose (n,k,1,0) ; ::_thesis: ex F being XFinSequence of st
( F = x & dom F = n & rng F c= {0,1} & Sum F = k )
then consider F being Function of n,{0,1} such that
A1: ( x = F & card (F " {1}) = k ) by Def1;
A2: rng F c= {0,1} ;
dom F = n by FUNCT_2:def_1;
then reconsider F = F as XFinSequence by AFINSQ_1:5;
rng F is Subset of NAT by A2, XBOOLE_1:1;
then reconsider F = F as XFinSequence of by RELAT_1:def_19;
take F ; ::_thesis: ( F = x & dom F = n & rng F c= {0,1} & Sum F = k )
Sum F = 1 * (card (F " {1})) by A2, AFINSQ_2:68;
hence ( F = x & dom F = n & rng F c= {0,1} & Sum F = k ) by A1, A2, FUNCT_2:def_1; ::_thesis: verum
end;
given F being XFinSequence of such that A3: F = x and
A4: ( dom F = n & rng F c= {0,1} & Sum F = k ) ; ::_thesis: x in Choose (n,k,1,0)
( 1 * (card (F " {1})) = k & F is Function of n,{0,1} ) by A4, AFINSQ_2:68, FUNCT_2:2;
hence x in Choose (n,k,1,0) by A3, Def1; ::_thesis: verum
end;
Lm2: for X being finite set ex P being Function of (card X),X st P is one-to-one
proof
let X be finite set ; ::_thesis: ex P being Function of (card X),X st P is one-to-one
card X,X are_equipotent by CARD_1:def_2;
then consider P being Function such that
A1: P is one-to-one and
A2: ( dom P = card X & rng P = X ) by WELLORD2:def_4;
P is Function of (card X),X by A2, FUNCT_2:1;
hence ex P being Function of (card X),X st P is one-to-one by A1; ::_thesis: verum
end;
definition
canceled;
let k be Nat;
let F be finite-yielding Function;
assume A1: dom F is finite ;
func Card_Intersection (F,k) -> Element of NAT means :Def4: :: CARD_FIN:def 4
for x, y being set
for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & it = Sum XFS );
existence
ex b1 being Element of NAT st
for x, y being set
for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & b1 = Sum XFS )
proof
reconsider D = dom F as finite set by A1;
set Ch1 = Choose (D,k,0,1);
card (Choose (D,k,0,1)), Choose (D,k,0,1) are_equipotent by CARD_1:def_2;
then consider P1 being Function such that
A2: P1 is one-to-one and
A3: dom P1 = card (Choose (D,k,0,1)) and
A4: rng P1 = Choose (D,k,0,1) by WELLORD2:def_4;
reconsider P1 = P1 as Function of (card (Choose (D,k,0,1))),(Choose (D,k,0,1)) by A3, A4, FUNCT_2:1;
defpred S1[ set , set ] means for f being Function st f = P1 . $1 holds
$2 = card (Intersection (F,f,0));
A5: for x being set st x in card (Choose (D,k,0,1)) holds
ex y being set st
( y in NAT & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in card (Choose (D,k,0,1)) implies ex y being set st
( y in NAT & S1[x,y] ) )
assume x in card (Choose (D,k,0,1)) ; ::_thesis: ex y being set st
( y in NAT & S1[x,y] )
then x in dom P1 by CARD_1:27, FUNCT_2:def_1;
then P1 . x in rng P1 by FUNCT_1:def_3;
then consider f being Function of D,{0,1} such that
A6: f = P1 . x and
card (f " {0}) = k by Def1;
union (rng F) is finite by A1, Th39;
then reconsider I = Intersection (F,f,0) as finite set ;
take card I ; ::_thesis: ( card I in NAT & S1[x, card I] )
thus ( card I in NAT & S1[x, card I] ) by A6; ::_thesis: verum
end;
consider XFS1 being Function of (card (Choose (D,k,0,1))),NAT such that
A7: for x being set st x in card (Choose (D,k,0,1)) holds
S1[x,XFS1 . x] from FUNCT_2:sch_1(A5);
A8: dom XFS1 = card (Choose (D,k,0,1)) by FUNCT_2:def_1;
then reconsider XFS1 = XFS1 as XFinSequence by AFINSQ_1:5;
reconsider XFS1 = XFS1 as XFinSequence of ;
reconsider S = Sum XFS1 as Element of NAT by ORDINAL1:def_12;
take S ; ::_thesis: for x, y being set
for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & S = Sum XFS )
let x, y be set ; ::_thesis: for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & S = Sum XFS )
let X be finite set ; ::_thesis: for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & S = Sum XFS )
let P be Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)); ::_thesis: ( dom F = X & P is one-to-one & x <> y implies ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & S = Sum XFS ) )
assume that
A9: dom F = X and
A10: P is one-to-one and
A11: x <> y ; ::_thesis: ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & S = Sum XFS )
defpred S2[ set , set ] means for f1 being Function of D,{0,1}
for f being Function of X,{x,y} st f1 = P1 . $1 & f = P . $2 holds
( f1 " {0} = f " {x} & ( for z being set st z in X holds
( ( f1 . z = 0 implies f . z = x ) & ( f . z = x implies f1 . z = 0 ) & ( f1 . z = 1 implies f . z = y ) & ( f . z = y implies f1 . z = 1 ) ) ) );
set Ch = Choose (X,k,x,y);
A12: for x1 being set st x1 in card (Choose (D,k,0,1)) holds
ex x2 being set st
( x2 in card (Choose (D,k,0,1)) & S2[x1,x2] )
proof
card (card (Choose (X,k,x,y))) = card (Choose (X,k,x,y)) ;
then P is onto by A10, STIRL2_1:60;
then A13: rng P = Choose (X,k,x,y) by FUNCT_2:def_3;
let x1 be set ; ::_thesis: ( x1 in card (Choose (D,k,0,1)) implies ex x2 being set st
( x2 in card (Choose (D,k,0,1)) & S2[x1,x2] ) )
assume x1 in card (Choose (D,k,0,1)) ; ::_thesis: ex x2 being set st
( x2 in card (Choose (D,k,0,1)) & S2[x1,x2] )
then P1 . x1 in rng P1 by A3, FUNCT_1:def_3;
then consider f1 being Function of D,{0,1} such that
A14: f1 = P1 . x1 and
A15: card (f1 " {0}) = k by Def1;
defpred S3[ set , set ] means ( ( f1 . $1 = 0 implies $2 = x ) & ( $2 = x implies f1 . $1 = 0 ) & ( f1 . $1 = 1 implies $2 = y ) & ( $2 = y implies f1 . $1 = 1 ) );
A16: for d being set st d in X holds
ex fd being set st
( fd in {x,y} & S3[d,fd] )
proof
let d be set ; ::_thesis: ( d in X implies ex fd being set st
( fd in {x,y} & S3[d,fd] ) )
assume d in X ; ::_thesis: ex fd being set st
( fd in {x,y} & S3[d,fd] )
then d in dom f1 by A9, FUNCT_2:def_1;
then f1 . d in rng f1 by FUNCT_1:def_3;
then A17: ( f1 . d = 0 or f1 . d = 1 ) by TARSKI:def_2;
( x in {x,y} & y in {x,y} ) by TARSKI:def_2;
hence ex fd being set st
( fd in {x,y} & S3[d,fd] ) by A11, A17; ::_thesis: verum
end;
consider f being Function of X,{x,y} such that
A18: for d being set st d in X holds
S3[d,f . d] from FUNCT_2:sch_1(A16);
A19: dom f1 = D by FUNCT_2:def_1;
A20: f1 " {0} c= f " {x}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in f1 " {0} or z in f " {x} )
assume A21: z in f1 " {0} ; ::_thesis: z in f " {x}
then f1 . z in {0} by FUNCT_1:def_7;
then A22: f1 . z = 0 by TARSKI:def_1;
A23: dom f1 = dom f by A9, A19, FUNCT_2:def_1;
then z in dom f by A19, A21;
then f . z = x by A18, A22;
then f . z in {x} by TARSKI:def_1;
hence z in f " {x} by A19, A21, A23, FUNCT_1:def_7; ::_thesis: verum
end;
A24: f " {x} c= f1 " {0}
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in f " {x} or z in f1 " {0} )
assume A25: z in f " {x} ; ::_thesis: z in f1 " {0}
then f . z in {x} by FUNCT_1:def_7;
then f . z = x by TARSKI:def_1;
then f1 . z = 0 by A18, A25;
then f1 . z in {0} by TARSKI:def_1;
hence z in f1 " {0} by A9, A19, A25, FUNCT_1:def_7; ::_thesis: verum
end;
then f " {x} = f1 " {0} by A20, XBOOLE_0:def_10;
then f in Choose (X,k,x,y) by A15, Def1;
then consider x2 being set such that
A26: x2 in dom P and
A27: P . x2 = f by A13, FUNCT_1:def_3;
take x2 ; ::_thesis: ( x2 in card (Choose (D,k,0,1)) & S2[x1,x2] )
( card (Choose (X,k,x,y)) = (card X) choose k & card (Choose (D,k,0,1)) = (card D) choose k ) by A11, Th16;
hence x2 in card (Choose (D,k,0,1)) by A9, A26; ::_thesis: S2[x1,x2]
let f19 be Function of D,{0,1}; ::_thesis: for f being Function of X,{x,y} st f19 = P1 . x1 & f = P . x2 holds
( f19 " {0} = f " {x} & ( for z being set st z in X holds
( ( f19 . z = 0 implies f . z = x ) & ( f . z = x implies f19 . z = 0 ) & ( f19 . z = 1 implies f . z = y ) & ( f . z = y implies f19 . z = 1 ) ) ) )
let f9 be Function of X,{x,y}; ::_thesis: ( f19 = P1 . x1 & f9 = P . x2 implies ( f19 " {0} = f9 " {x} & ( for z being set st z in X holds
( ( f19 . z = 0 implies f9 . z = x ) & ( f9 . z = x implies f19 . z = 0 ) & ( f19 . z = 1 implies f9 . z = y ) & ( f9 . z = y implies f19 . z = 1 ) ) ) ) )
assume A28: ( f19 = P1 . x1 & f9 = P . x2 ) ; ::_thesis: ( f19 " {0} = f9 " {x} & ( for z being set st z in X holds
( ( f19 . z = 0 implies f9 . z = x ) & ( f9 . z = x implies f19 . z = 0 ) & ( f19 . z = 1 implies f9 . z = y ) & ( f9 . z = y implies f19 . z = 1 ) ) ) )
thus f9 " {x} = f19 " {0} by A14, A24, A20, A27, A28, XBOOLE_0:def_10; ::_thesis: for z being set st z in X holds
( ( f19 . z = 0 implies f9 . z = x ) & ( f9 . z = x implies f19 . z = 0 ) & ( f19 . z = 1 implies f9 . z = y ) & ( f9 . z = y implies f19 . z = 1 ) )
let z be set ; ::_thesis: ( z in X implies ( ( f19 . z = 0 implies f9 . z = x ) & ( f9 . z = x implies f19 . z = 0 ) & ( f19 . z = 1 implies f9 . z = y ) & ( f9 . z = y implies f19 . z = 1 ) ) )
assume z in X ; ::_thesis: ( ( f19 . z = 0 implies f9 . z = x ) & ( f9 . z = x implies f19 . z = 0 ) & ( f19 . z = 1 implies f9 . z = y ) & ( f9 . z = y implies f19 . z = 1 ) )
hence ( ( f19 . z = 0 implies f9 . z = x ) & ( f9 . z = x implies f19 . z = 0 ) & ( f19 . z = 1 implies f9 . z = y ) & ( f9 . z = y implies f19 . z = 1 ) ) by A14, A18, A27, A28; ::_thesis: verum
end;
consider Perm being Function of (card (Choose (D,k,0,1))),(card (Choose (D,k,0,1))) such that
A29: for x1 being set st x1 in card (Choose (D,k,0,1)) holds
S2[x1,Perm . x1] from FUNCT_2:sch_1(A12);
now__::_thesis:_for_z1,_z2_being_set_st_z1_in_dom_Perm_&_z2_in_dom_Perm_&_Perm_._z1_=_Perm_._z2_holds_
z1_=_z2
A30: ( Choose (X,k,x,y) = {} implies card (Choose (X,k,x,y)) = {} ) ;
let z1, z2 be set ; ::_thesis: ( z1 in dom Perm & z2 in dom Perm & Perm . z1 = Perm . z2 implies z1 = z2 )
assume that
A31: z1 in dom Perm and
A32: z2 in dom Perm and
A33: Perm . z1 = Perm . z2 ; ::_thesis: z1 = z2
( card (Choose (X,k,x,y)) = (card X) choose k & card (Choose (D,k,0,1)) = (card D) choose k ) by A11, Th16;
then Perm . z1 in card (Choose (X,k,x,y)) by A9, A31;
then Perm . z1 in dom P by A30, FUNCT_2:def_1;
then P . (Perm . z1) in rng P by FUNCT_1:def_3;
then consider PPermz1 being Function of X,{x,y} such that
A34: PPermz1 = P . (Perm . z1) and
card (PPermz1 " {x}) = k by Def1;
P1 . z2 in rng P1 by A3, A32, FUNCT_1:def_3;
then consider P1z2 being Function of D,{0,1} such that
A35: P1 . z2 = P1z2 and
card (P1z2 " {0}) = k by Def1;
P1 . z1 in rng P1 by A3, A31, FUNCT_1:def_3;
then consider P1z1 being Function of D,{0,1} such that
A36: P1 . z1 = P1z1 and
card (P1z1 " {0}) = k by Def1;
A37: for z being set st z in dom P1z1 holds
P1z1 . z = P1z2 . z
proof
let z be set ; ::_thesis: ( z in dom P1z1 implies P1z1 . z = P1z2 . z )
assume A38: z in dom P1z1 ; ::_thesis: P1z1 . z = P1z2 . z
A39: P1z1 . z in rng P1z1 by A38, FUNCT_1:def_3;
percases ( P1z1 . z = 0 or P1z1 . z = 1 ) by A39, TARSKI:def_2;
supposeA40: P1z1 . z = 0 ; ::_thesis: P1z1 . z = P1z2 . z
then PPermz1 . z = x by A9, A29, A31, A36, A34, A38;
hence P1z1 . z = P1z2 . z by A9, A29, A32, A33, A35, A34, A38, A40; ::_thesis: verum
end;
supposeA41: P1z1 . z = 1 ; ::_thesis: P1z1 . z = P1z2 . z
then PPermz1 . z = y by A9, A29, A31, A36, A34, A38;
hence P1z1 . z = P1z2 . z by A9, A29, A32, A33, A35, A34, A38, A41; ::_thesis: verum
end;
end;
end;
( dom P1z1 = D & dom P1z2 = D ) by FUNCT_2:def_1;
then P1z1 = P1z2 by A37, FUNCT_1:def_11;
hence z1 = z2 by A2, A3, A31, A32, A36, A35, FUNCT_1:def_4; ::_thesis: verum
end;
then A42: Perm is one-to-one by FUNCT_1:def_4;
card (card (Choose (D,k,0,1))) = card (card (Choose (D,k,0,1))) ;
then A43: ( Perm is one-to-one & Perm is onto ) by A42, STIRL2_1:60;
defpred S3[ set , set ] means for f being Function st f = P . $1 holds
$2 = card (Intersection (F,f,x));
A44: for x1 being set st x1 in card (Choose (X,k,x,y)) holds
ex x2 being set st
( x2 in NAT & S3[x1,x2] )
proof
let x1 be set ; ::_thesis: ( x1 in card (Choose (X,k,x,y)) implies ex x2 being set st
( x2 in NAT & S3[x1,x2] ) )
assume x1 in card (Choose (X,k,x,y)) ; ::_thesis: ex x2 being set st
( x2 in NAT & S3[x1,x2] )
then x1 in dom P by CARD_1:27, FUNCT_2:def_1;
then P . x1 in rng P by FUNCT_1:def_3;
then consider f being Function of X,{x,y} such that
A45: f = P . x1 and
card (f " {x}) = k by Def1;
union (rng F) is finite by A1, Th39;
then reconsider I = Intersection (F,f,x) as finite set ;
take card I ; ::_thesis: ( card I in NAT & S3[x1, card I] )
thus ( card I in NAT & S3[x1, card I] ) by A45; ::_thesis: verum
end;
consider XFS being Function of (card (Choose (X,k,x,y))),NAT such that
A46: for x1 being set st x1 in card (Choose (X,k,x,y)) holds
S3[x1,XFS . x1] from FUNCT_2:sch_1(A44);
A47: dom XFS = card (Choose (X,k,x,y)) by FUNCT_2:def_1;
then reconsider XFS = XFS as XFinSequence by AFINSQ_1:5;
reconsider XFS = XFS as XFinSequence of ;
take XFS ; ::_thesis: ( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & S = Sum XFS )
( Choose (X,k,x,y) = {} implies card (Choose (X,k,x,y)) = {} ) ;
then dom P = card (Choose (X,k,x,y)) by FUNCT_2:def_1;
hence A48: dom XFS = dom P by FUNCT_2:def_1; ::_thesis: ( ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & S = Sum XFS )
hence for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) by A46; ::_thesis: S = Sum XFS
A49: card (Choose (D,k,0,1)) = (card D) choose k by Th16;
A50: card (Choose (X,k,x,y)) = (card X) choose k by A11, Th16;
then card (Choose (D,k,0,1)) = dom XFS by A9, A49, FUNCT_2:def_1;
then reconsider Perm = Perm as Permutation of (dom XFS) by A43;
A51: dom XFS = dom XFS1 by A9, A47, A49, A50, FUNCT_2:def_1;
A52: for z being set st z in dom XFS1 holds
XFS1 . z = (XFS * Perm) . z
proof
let z be set ; ::_thesis: ( z in dom XFS1 implies XFS1 . z = (XFS * Perm) . z )
assume A53: z in dom XFS1 ; ::_thesis: XFS1 . z = (XFS * Perm) . z
A54: z in dom Perm by A8, A53, FUNCT_2:52;
P . (Perm . z) in rng P by A48, A51, A53, FUNCT_1:def_3;
then consider p being Function of X,{x,y} such that
A55: p = P . (Perm . z) and
card (p " {x}) = k by Def1;
A56: XFS . (Perm . z) = card (Intersection (F,p,x)) by A46, A47, A51, A53, A55;
P1 . z in rng P1 by A3, A8, A53, FUNCT_1:def_3;
then consider p1 being Function of D,{0,1} such that
A57: p1 = P1 . z and
card (p1 " {0}) = k by Def1;
p1 " {0} = p " {x} by A8, A29, A53, A55, A57;
then A58: Intersection (F,p1,0) = Intersection (F,p,x) by Th33;
XFS1 . z = card (Intersection (F,p1,0)) by A7, A8, A53, A57;
hence XFS1 . z = (XFS * Perm) . z by A58, A56, A54, FUNCT_1:13; ::_thesis: verum
end;
( rng Perm c= dom XFS & dom Perm = dom XFS ) by FUNCT_2:52;
then dom XFS1 = dom (XFS * Perm) by A51, RELAT_1:27;
then XFS1 = XFS * Perm by A52, FUNCT_1:def_11;
then A59: addnat "**" XFS = addnat "**" XFS1 by AFINSQ_2:45;
addnat "**" XFS1 = Sum XFS1 by AFINSQ_2:51;
hence S = Sum XFS by A59, AFINSQ_2:51; ::_thesis: verum
end;
uniqueness
for b1, b2 being Element of NAT st ( for x, y being set
for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & b1 = Sum XFS ) ) & ( for x, y being set
for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & b2 = Sum XFS ) ) holds
b1 = b2
proof
reconsider D = dom F as finite set by A1;
let n1, n2 be Element of NAT ; ::_thesis: ( ( for x, y being set
for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & n1 = Sum XFS ) ) & ( for x, y being set
for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & n2 = Sum XFS ) ) implies n1 = n2 )
assume that
A60: for x, y being set
for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & n1 = Sum XFS ) and
A61: for x, y being set
for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & n2 = Sum XFS ) ; ::_thesis: n1 = n2
set Ch1 = Choose (D,k,0,1);
card (Choose (D,k,0,1)), Choose (D,k,0,1) are_equipotent by CARD_1:def_2;
then consider P being Function such that
A62: P is one-to-one and
A63: ( dom P = card (Choose (D,k,0,1)) & rng P = Choose (D,k,0,1) ) by WELLORD2:def_4;
reconsider P = P as Function of (card (Choose (D,k,0,1))),(Choose (D,k,0,1)) by A63, FUNCT_2:1;
consider XFS1 being XFinSequence of such that
A64: dom XFS1 = dom P and
A65: for z being set
for f being Function st z in dom XFS1 & f = P . z holds
XFS1 . z = card (Intersection (F,f,0)) and
A66: n1 = Sum XFS1 by A60, A62;
consider XFS2 being XFinSequence of such that
A67: dom XFS2 = dom P and
A68: for z being set
for f being Function st z in dom XFS2 & f = P . z holds
XFS2 . z = card (Intersection (F,f,0)) and
A69: n2 = Sum XFS2 by A61, A62;
now__::_thesis:_for_z_being_set_st_z_in_dom_XFS1_holds_
XFS2_._z_=_XFS1_._z
let z be set ; ::_thesis: ( z in dom XFS1 implies XFS2 . z = XFS1 . z )
assume A70: z in dom XFS1 ; ::_thesis: XFS2 . z = XFS1 . z
P . z in rng P by A64, A70, FUNCT_1:def_3;
then consider Pz being Function of D,{0,1} such that
A71: Pz = P . z and
card (Pz " {0}) = k by Def1;
XFS2 . z = card (Intersection (F,Pz,0)) by A64, A67, A68, A70, A71;
hence XFS2 . z = XFS1 . z by A65, A70, A71; ::_thesis: verum
end;
hence n1 = n2 by A64, A66, A67, A69, FUNCT_1:2; ::_thesis: verum
end;
end;
:: deftheorem CARD_FIN:def_3_:_
canceled;
:: deftheorem Def4 defines Card_Intersection CARD_FIN:def_4_:_
for k being Nat
for F being finite-yielding Function st dom F is finite holds
for b3 being Element of NAT holds
( b3 = Card_Intersection (F,k) iff for x, y being set
for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom F = X & P is one-to-one & x <> y holds
ex XFS being XFinSequence of st
( dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (F,f,x)) ) & b3 = Sum XFS ) );
theorem :: CARD_FIN:41
for k being Nat
for Fy being finite-yielding Function
for x, y being set
for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom Fy = X & P is one-to-one & x <> y holds
for XFS being XFinSequence of st dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (Fy,f,x)) ) holds
Card_Intersection (Fy,k) = Sum XFS
proof
let k be Nat; ::_thesis: for Fy being finite-yielding Function
for x, y being set
for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom Fy = X & P is one-to-one & x <> y holds
for XFS being XFinSequence of st dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (Fy,f,x)) ) holds
Card_Intersection (Fy,k) = Sum XFS
let Fy be finite-yielding Function; ::_thesis: for x, y being set
for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom Fy = X & P is one-to-one & x <> y holds
for XFS being XFinSequence of st dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (Fy,f,x)) ) holds
Card_Intersection (Fy,k) = Sum XFS
let x, y be set ; ::_thesis: for X being finite set
for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom Fy = X & P is one-to-one & x <> y holds
for XFS being XFinSequence of st dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (Fy,f,x)) ) holds
Card_Intersection (Fy,k) = Sum XFS
let X be finite set ; ::_thesis: for P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) st dom Fy = X & P is one-to-one & x <> y holds
for XFS being XFinSequence of st dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (Fy,f,x)) ) holds
Card_Intersection (Fy,k) = Sum XFS
let P be Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)); ::_thesis: ( dom Fy = X & P is one-to-one & x <> y implies for XFS being XFinSequence of st dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (Fy,f,x)) ) holds
Card_Intersection (Fy,k) = Sum XFS )
assume ( dom Fy = X & P is one-to-one & x <> y ) ; ::_thesis: for XFS being XFinSequence of st dom XFS = dom P & ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (Fy,f,x)) ) holds
Card_Intersection (Fy,k) = Sum XFS
then consider XFS being XFinSequence of such that
A1: dom XFS = dom P and
A2: for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (Fy,f,x)) and
A3: Card_Intersection (Fy,k) = Sum XFS by Def4;
let XFS1 be XFinSequence of ; ::_thesis: ( dom XFS1 = dom P & ( for z being set
for f being Function st z in dom XFS1 & f = P . z holds
XFS1 . z = card (Intersection (Fy,f,x)) ) implies Card_Intersection (Fy,k) = Sum XFS1 )
assume that
A4: dom XFS1 = dom P and
A5: for z being set
for f being Function st z in dom XFS1 & f = P . z holds
XFS1 . z = card (Intersection (Fy,f,x)) ; ::_thesis: Card_Intersection (Fy,k) = Sum XFS1
now__::_thesis:_for_z_being_set_st_z_in_dom_XFS_holds_
XFS1_._z_=_XFS_._z
let z be set ; ::_thesis: ( z in dom XFS implies XFS1 . z = XFS . z )
assume A6: z in dom XFS ; ::_thesis: XFS1 . z = XFS . z
P . z in rng P by A1, A6, FUNCT_1:def_3;
then consider Pz being Function of X,{x,y} such that
A7: Pz = P . z and
card (Pz " {x}) = k by Def1;
XFS1 . z = card (Intersection (Fy,Pz,x)) by A4, A5, A1, A6, A7;
hence XFS1 . z = XFS . z by A2, A6, A7; ::_thesis: verum
end;
hence Card_Intersection (Fy,k) = Sum XFS1 by A4, A1, A3, FUNCT_1:2; ::_thesis: verum
end;
theorem :: CARD_FIN:42
for k being Nat
for Fy being finite-yielding Function st dom Fy is finite & k = 0 holds
Card_Intersection (Fy,k) = card (union (rng Fy))
proof
let k be Nat; ::_thesis: for Fy being finite-yielding Function st dom Fy is finite & k = 0 holds
Card_Intersection (Fy,k) = card (union (rng Fy))
let Fy be finite-yielding Function; ::_thesis: ( dom Fy is finite & k = 0 implies Card_Intersection (Fy,k) = card (union (rng Fy)) )
assume that
A1: dom Fy is finite and
A2: k = 0 ; ::_thesis: Card_Intersection (Fy,k) = card (union (rng Fy))
reconsider X = dom Fy as finite set by A1;
set Ch = Choose (X,k,0,1);
consider P being Function of (card (Choose (X,k,0,1))),(Choose (X,k,0,1)) such that
A3: P is one-to-one by Lm2;
A4: card (Choose (X,k,0,1)) = 1 by A2, Th11;
then A5: dom P = 1 by CARD_1:27, FUNCT_2:def_1;
consider XFS being XFinSequence of such that
A6: dom XFS = dom P and
A7: for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (Fy,f,0)) and
A8: Card_Intersection (Fy,k) = Sum XFS by A3, Def4;
len XFS = 1 by A6, A4, CARD_1:27, FUNCT_2:def_1;
then XFS = <%(XFS . 0)%> by AFINSQ_1:34;
then A9: addnat "**" XFS = XFS . 0 by AFINSQ_2:37;
A10: 0 in 1 by CARD_1:49, TARSKI:def_1;
then P . 0 in rng P by A5, FUNCT_1:def_3;
then consider P0 being Function of X,{0,1} such that
A11: P0 = P . 0 and
A12: card (P0 " {0}) = 0 by A2, Def1;
P0 " {0} = {} by A12;
then A13: Intersection (Fy,P0,0) = union (rng Fy) by Th34;
XFS . 0 = card (Intersection (Fy,P0,0)) by A6, A7, A5, A10, A11;
hence Card_Intersection (Fy,k) = card (union (rng Fy)) by A8, A13, A9, AFINSQ_2:51; ::_thesis: verum
end;
theorem Th43: :: CARD_FIN:43
for X being finite set
for k being Nat
for Fy being finite-yielding Function st dom Fy = X & k > card X holds
Card_Intersection (Fy,k) = 0
proof
let X be finite set ; ::_thesis: for k being Nat
for Fy being finite-yielding Function st dom Fy = X & k > card X holds
Card_Intersection (Fy,k) = 0
let k be Nat; ::_thesis: for Fy being finite-yielding Function st dom Fy = X & k > card X holds
Card_Intersection (Fy,k) = 0
let Fy be finite-yielding Function; ::_thesis: ( dom Fy = X & k > card X implies Card_Intersection (Fy,k) = 0 )
assume that
A1: dom Fy = X and
A2: k > card X ; ::_thesis: Card_Intersection (Fy,k) = 0
set Ch = Choose (X,k,0,1);
consider P being Function of (card (Choose (X,k,0,1))),(Choose (X,k,0,1)) such that
A3: P is one-to-one by Lm2;
consider XFS being XFinSequence of such that
A4: dom XFS = dom P and
for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (Fy,f,0)) and
A5: Card_Intersection (Fy,k) = Sum XFS by A1, A3, Def4;
Choose (X,k,0,1) is empty by A2, Th10;
then XFS = 0 by A4;
hence Card_Intersection (Fy,k) = 0 by A5; ::_thesis: verum
end;
theorem Th44: :: CARD_FIN:44
for Fy being finite-yielding Function
for X being finite set st dom Fy = X holds
for P being Function of (card X),X st P is one-to-one holds
ex XFS being XFinSequence of st
( dom XFS = card X & ( for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) ) & Card_Intersection (Fy,1) = Sum XFS )
proof
let Fy be finite-yielding Function; ::_thesis: for X being finite set st dom Fy = X holds
for P being Function of (card X),X st P is one-to-one holds
ex XFS being XFinSequence of st
( dom XFS = card X & ( for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) ) & Card_Intersection (Fy,1) = Sum XFS )
let X be finite set ; ::_thesis: ( dom Fy = X implies for P being Function of (card X),X st P is one-to-one holds
ex XFS being XFinSequence of st
( dom XFS = card X & ( for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) ) & Card_Intersection (Fy,1) = Sum XFS ) )
assume A1: dom Fy = X ; ::_thesis: for P being Function of (card X),X st P is one-to-one holds
ex XFS being XFinSequence of st
( dom XFS = card X & ( for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) ) & Card_Intersection (Fy,1) = Sum XFS )
let P be Function of (card X),X; ::_thesis: ( P is one-to-one implies ex XFS being XFinSequence of st
( dom XFS = card X & ( for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) ) & Card_Intersection (Fy,1) = Sum XFS ) )
assume A2: P is one-to-one ; ::_thesis: ex XFS being XFinSequence of st
( dom XFS = card X & ( for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) ) & Card_Intersection (Fy,1) = Sum XFS )
percases ( X = {} or X <> {} ) ;
supposeA3: X = {} ; ::_thesis: ex XFS being XFinSequence of st
( dom XFS = card X & ( for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) ) & Card_Intersection (Fy,1) = Sum XFS )
reconsider XFS = {} as XFinSequence ;
( rng {} c= {} & {} c= NAT ) ;
then reconsider XFS = XFS as XFinSequence of by RELAT_1:def_19;
take XFS ; ::_thesis: ( dom XFS = card X & ( for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) ) & Card_Intersection (Fy,1) = Sum XFS )
thus ( card X = dom XFS & ( for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) ) ) by A3, CARD_1:27; ::_thesis: Card_Intersection (Fy,1) = Sum XFS
Sum XFS = 0 ;
hence Card_Intersection (Fy,1) = Sum XFS by A1, A3, Th43, CARD_1:27; ::_thesis: verum
end;
suppose X <> {} ; ::_thesis: ex XFS being XFinSequence of st
( dom XFS = card X & ( for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) ) & Card_Intersection (Fy,1) = Sum XFS )
then reconsider cX = card X as non empty set ;
deffunc H1( Element of cX) -> Element of NAT = card ((Fy * P) . $1);
consider XFS being Function of cX,NAT such that
A4: for x being Element of cX holds XFS . x = H1(x) from FUNCT_2:sch_4();
A5: dom XFS = cX by FUNCT_2:def_1;
then reconsider XFS = XFS as XFinSequence by AFINSQ_1:5;
reconsider XFS = XFS as XFinSequence of ;
take XFS ; ::_thesis: ( dom XFS = card X & ( for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) ) & Card_Intersection (Fy,1) = Sum XFS )
thus card X = dom XFS by FUNCT_2:def_1; ::_thesis: ( ( for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) ) & Card_Intersection (Fy,1) = Sum XFS )
thus for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) by A4, A5; ::_thesis: Card_Intersection (Fy,1) = Sum XFS
thus Card_Intersection (Fy,1) = Sum XFS ::_thesis: verum
proof
deffunc H2( set ) -> set = ((P . $1) .--> 0) +* ((X \ {(P . $1)}) --> 1);
A6: for x being set st x in cX holds
H2(x) in Choose (X,1,0,1)
proof
let x be set ; ::_thesis: ( x in cX implies H2(x) in Choose (X,1,0,1) )
assume x in cX ; ::_thesis: H2(x) in Choose (X,1,0,1)
then x in dom P by CARD_1:27, FUNCT_2:def_1;
then P . x in rng P by FUNCT_1:def_3;
then A7: {(P . x)} \/ (X \ {(P . x)}) = X by ZFMISC_1:116;
( {(P . x)} misses X \ {(P . x)} & card {(P . x)} = 1 ) by CARD_1:30, XBOOLE_1:79;
hence H2(x) in Choose (X,1,0,1) by A7, Th18; ::_thesis: verum
end;
consider P1 being Function of cX,(Choose (X,1,0,1)) such that
A8: for z being set st z in cX holds
P1 . z = H2(z) from FUNCT_2:sch_2(A6);
Choose (X,1,0,1) c= rng P1
proof
card X = card (card X) ;
then A9: P is onto by A2, STIRL2_1:60;
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Choose (X,1,0,1) or z in rng P1 )
assume z in Choose (X,1,0,1) ; ::_thesis: z in rng P1
then consider F being Function of X,{0,1} such that
A10: F = z and
A11: card (F " {0}) = 1 by Def1;
consider x1 being set such that
A12: F " {0} = {x1} by A11, CARD_2:42;
A13: x1 in {x1} by TARSKI:def_1;
then x1 in X by A12;
then x1 in rng P by A9, FUNCT_2:def_3;
then consider x2 being set such that
A14: x2 in dom P and
A15: P . x2 = x1 by FUNCT_1:def_3;
A16: P1 . x2 = F
proof
set F1 = (X \ {(P . x2)}) --> 1;
set F0 = (P . x2) .--> 0;
set P1x = ((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1);
A17: {(P . x2)} \/ (X \ {(P . x2)}) = X by A12, A13, A15, ZFMISC_1:116;
A18: now__::_thesis:_for_d_being_set_st_d_in_dom_F_holds_
(((P_._x2)_.-->_0)_+*_((X_\_{(P_._x2)})_-->_1))_._d_=_F_._d
let d be set ; ::_thesis: ( d in dom F implies (((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1)) . d = F . d )
assume A19: d in dom F ; ::_thesis: (((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1)) . d = F . d
now__::_thesis:_(((P_._x2)_.-->_0)_+*_((X_\_{(P_._x2)})_-->_1))_._d_=_F_._d
percases ( d in {(P . x2)} or d in X \ {(P . x2)} ) by A17, A19, XBOOLE_0:def_3;
supposeA20: d in {(P . x2)} ; ::_thesis: (((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1)) . d = F . d
A21: {(P . x2)} misses X \ {(P . x2)} by XBOOLE_1:79;
( dom ((P . x2) .--> 0) = {(P . x2)} & dom ((X \ {(P . x2)}) --> 1) = X \ {(P . x2)} ) by FUNCOP_1:13;
then (((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1)) . d = ((P . x2) .--> 0) . d by A20, A21, FUNCT_4:16;
then A22: (((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1)) . d = 0 by A20, FUNCOP_1:7;
F . d in {0} by A12, A15, A20, FUNCT_1:def_7;
hence (((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1)) . d = F . d by A22, TARSKI:def_1; ::_thesis: verum
end;
supposeA23: d in X \ {(P . x2)} ; ::_thesis: (((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1)) . d = F . d
then d in dom ((X \ {(P . x2)}) --> 1) by FUNCT_2:def_1;
then (((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1)) . d = ((X \ {(P . x2)}) --> 1) . d by FUNCT_4:13;
then A24: (((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1)) . d = 1 by A23, FUNCOP_1:7;
A25: X = dom F by FUNCT_2:def_1;
not d in {x1} by A15, A23, XBOOLE_0:def_5;
then not F . d in {0} by A12, A23, A25, FUNCT_1:def_7;
then A26: not F . d = 0 by TARSKI:def_1;
F . d in rng F by A23, A25, FUNCT_1:def_3;
hence (((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1)) . d = F . d by A24, A26, TARSKI:def_2; ::_thesis: verum
end;
end;
end;
hence (((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1)) . d = F . d ; ::_thesis: verum
end;
dom ((P . x2) .--> 0) = {(P . x2)} by FUNCOP_1:13;
then A27: X = (dom ((P . x2) .--> 0)) \/ (dom ((X \ {(P . x2)}) --> 1)) by A17, FUNCT_2:def_1;
( dom F = X & dom (((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1)) = (dom ((P . x2) .--> 0)) \/ (dom ((X \ {(P . x2)}) --> 1)) ) by FUNCT_2:def_1, FUNCT_4:def_1;
then ((P . x2) .--> 0) +* ((X \ {(P . x2)}) --> 1) = F by A27, A18, FUNCT_1:2;
hence P1 . x2 = F by A8, A14; ::_thesis: verum
end;
card (Choose (X,1,0,1)) = (card X) choose 1 by Th16;
then card (Choose (X,1,0,1)) = cX by NAT_1:14, NEWTON:23;
then dom P1 = cX by CARD_1:27, FUNCT_2:def_1;
hence z in rng P1 by A10, A14, A16, FUNCT_1:def_3; ::_thesis: verum
end;
then A28: Choose (X,1,0,1) = rng P1 by XBOOLE_0:def_10;
then A29: P1 is onto by FUNCT_2:def_3;
card (Choose (X,1,0,1)) = (card X) choose 1 by Th16;
then A30: card X = card (Choose (X,1,0,1)) by A28, NAT_1:14, NEWTON:23;
then reconsider P1 = P1 as Function of (card (Choose (X,1,0,1))),(Choose (X,1,0,1)) ;
card (card X) = card X ;
then P1 is one-to-one by A29, A30, STIRL2_1:60;
then consider XFS1 being XFinSequence of such that
A31: dom XFS1 = dom P1 and
A32: for z being set
for f being Function st z in dom XFS1 & f = P1 . z holds
XFS1 . z = card (Intersection (Fy,f,0)) and
A33: Card_Intersection (Fy,1) = Sum XFS1 by A1, Def4;
( Choose (X,1,0,1) = {} implies card (Choose (X,1,0,1)) = {} ) ;
then A34: dom P1 = card (Choose (X,1,0,1)) by FUNCT_2:def_1;
A35: for z being set st z in dom XFS1 holds
XFS1 . z = XFS . z
proof
let z be set ; ::_thesis: ( z in dom XFS1 implies XFS1 . z = XFS . z )
assume A36: z in dom XFS1 ; ::_thesis: XFS1 . z = XFS . z
H2(z) in Choose (X,1,0,1) by A6, A30, A31, A36;
then consider f being Function of X,{0,1} such that
A37: f = H2(z) and
A38: card (f " {0}) = 1 by Def1;
consider x1 being set such that
A39: f " {0} = {x1} by A38, CARD_2:42;
P1 . z = H2(z) by A8, A30, A31, A36;
then A40: XFS1 . z = card (Intersection (Fy,f,0)) by A32, A36, A37;
A41: 0 in {0} by TARSKI:def_1;
A42: dom ((X \ {(P . z)}) --> 1) = X \ {(P . z)} by FUNCOP_1:13;
A43: P . z in {(P . z)} by TARSKI:def_1;
{(P . z)} = dom ((P . z) .--> 0) by FUNCOP_1:13;
then A44: P . z in (dom ((P . z) .--> 0)) \/ (dom ((X \ {(P . z)}) --> 1)) by A43, XBOOLE_0:def_3;
( not P . z in X \ {(P . z)} & ((P . z) .--> 0) . (P . z) = 0 ) by A43, FUNCOP_1:7, XBOOLE_0:def_5;
then A45: H2(z) . (P . z) = 0 by A44, A42, FUNCT_4:def_1;
P . z in dom H2(z) by A44, FUNCT_4:def_1;
then A46: P . z in f " {0} by A37, A45, A41, FUNCT_1:def_7;
then P . z = x1 by A39, TARSKI:def_1;
then A47: card (Intersection (Fy,f,0)) = card (Fy . (P . z)) by A39, Th35;
A48: XFS . z = card ((Fy * P) . z) by A4, A30, A31, A36;
z in dom P by A30, A31, A34, A36, A46, FUNCT_2:def_1;
hence XFS1 . z = XFS . z by A47, A40, A48, FUNCT_1:13; ::_thesis: verum
end;
dom XFS1 = dom XFS by A30, A31, A34, FUNCT_2:def_1;
hence Card_Intersection (Fy,1) = Sum XFS by A33, A35, FUNCT_1:def_11; ::_thesis: verum
end;
end;
end;
end;
theorem Th45: :: CARD_FIN:45
for x being set
for X being finite set
for Fy being finite-yielding Function st dom Fy = X holds
Card_Intersection (Fy,(card X)) = card (Intersection (Fy,(X --> x),x))
proof
let x be set ; ::_thesis: for X being finite set
for Fy being finite-yielding Function st dom Fy = X holds
Card_Intersection (Fy,(card X)) = card (Intersection (Fy,(X --> x),x))
let X be finite set ; ::_thesis: for Fy being finite-yielding Function st dom Fy = X holds
Card_Intersection (Fy,(card X)) = card (Intersection (Fy,(X --> x),x))
let Fy be finite-yielding Function; ::_thesis: ( dom Fy = X implies Card_Intersection (Fy,(card X)) = card (Intersection (Fy,(X --> x),x)) )
set Ch = Choose (X,(card X),x,{x});
consider P being Function of (card (Choose (X,(card X),x,{x}))),(Choose (X,(card X),x,{x})) such that
A1: P is one-to-one by Lm2;
x in {x} by TARSKI:def_1;
then A2: x <> {x} ;
assume dom Fy = X ; ::_thesis: Card_Intersection (Fy,(card X)) = card (Intersection (Fy,(X --> x),x))
then consider XFS being XFinSequence of such that
A3: dom XFS = dom P and
A4: ( ( for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (Fy,f,x)) ) & Card_Intersection (Fy,(card X)) = Sum XFS ) by A1, A2, Def4;
A5: card (Choose (X,(card X),x,{x})) = 1 by Th12;
then consider ch being set such that
A6: Choose (X,(card X),x,{x}) = {ch} by CARD_2:42;
x in {x} by TARSKI:def_1;
then ( X \/ {} = X & {x} <> x ) ;
then ({} --> {x}) +* (X --> x) in Choose (X,(card X),x,{x}) by Th17;
then {} +* (X --> x) in Choose (X,(card X),x,{x}) ;
then X --> x in Choose (X,(card X),x,{x}) ;
then A7: X --> x = ch by A6, TARSKI:def_1;
A8: ( Choose (X,(card X),x,{x}) = {} implies card (Choose (X,(card X),x,{x})) = {} ) ;
then A9: dom P = card (Choose (X,(card X),x,{x})) by FUNCT_2:def_1;
then 0 in dom P by A5, CARD_1:49, TARSKI:def_1;
then P . 0 in rng P by FUNCT_1:def_3;
then A10: P . 0 = ch by A6, TARSKI:def_1;
len XFS = 1 by A3, A8, A5, FUNCT_2:def_1;
then XFS = <%(XFS . 0)%> by AFINSQ_1:34;
then addnat "**" XFS = XFS . 0 by AFINSQ_2:37;
then A11: Sum XFS = XFS . 0 by AFINSQ_2:51;
0 in dom XFS by A3, A5, A9, CARD_1:49, TARSKI:def_1;
hence Card_Intersection (Fy,(card X)) = card (Intersection (Fy,(X --> x),x)) by A4, A11, A10, A7; ::_thesis: verum
end;
theorem Th46: :: CARD_FIN:46
for x being set
for X being finite set
for Fy being finite-yielding Function st Fy = x .--> X holds
Card_Intersection (Fy,1) = card X
proof
let x be set ; ::_thesis: for X being finite set
for Fy being finite-yielding Function st Fy = x .--> X holds
Card_Intersection (Fy,1) = card X
let X be finite set ; ::_thesis: for Fy being finite-yielding Function st Fy = x .--> X holds
Card_Intersection (Fy,1) = card X
let Fy be finite-yielding Function; ::_thesis: ( Fy = x .--> X implies Card_Intersection (Fy,1) = card X )
assume A1: Fy = x .--> X ; ::_thesis: Card_Intersection (Fy,1) = card X
then A2: dom Fy = {x} by FUNCOP_1:13;
A3: x in {x} by TARSKI:def_1;
then A4: (x .--> x) " {x} = {x} by FUNCOP_1:14;
Fy . x = X by A1, A3, FUNCOP_1:7;
then ( 1 = card {x} & Intersection (Fy,(x .--> x),x) = X ) by A4, Th35, CARD_1:30;
hence Card_Intersection (Fy,1) = card X by A2, Th45; ::_thesis: verum
end;
theorem :: CARD_FIN:47
for x, y being set
for X, Y being finite set
for Fy being finite-yielding Function st x <> y & Fy = (x,y) --> (X,Y) holds
( Card_Intersection (Fy,1) = (card X) + (card Y) & Card_Intersection (Fy,2) = card (X /\ Y) )
proof
let x, y be set ; ::_thesis: for X, Y being finite set
for Fy being finite-yielding Function st x <> y & Fy = (x,y) --> (X,Y) holds
( Card_Intersection (Fy,1) = (card X) + (card Y) & Card_Intersection (Fy,2) = card (X /\ Y) )
let X, Y be finite set ; ::_thesis: for Fy being finite-yielding Function st x <> y & Fy = (x,y) --> (X,Y) holds
( Card_Intersection (Fy,1) = (card X) + (card Y) & Card_Intersection (Fy,2) = card (X /\ Y) )
let Fy be finite-yielding Function; ::_thesis: ( x <> y & Fy = (x,y) --> (X,Y) implies ( Card_Intersection (Fy,1) = (card X) + (card Y) & Card_Intersection (Fy,2) = card (X /\ Y) ) )
assume that
A1: x <> y and
A2: Fy = (x,y) --> (X,Y) ; ::_thesis: ( Card_Intersection (Fy,1) = (card X) + (card Y) & Card_Intersection (Fy,2) = card (X /\ Y) )
set P = (0,1) --> (x,y);
A3: ( dom ((0,1) --> (x,y)) = {0,1} & rng ((0,1) --> (x,y)) = {x,y} ) by FUNCT_4:62, FUNCT_4:64;
card {x,y} = 2 by A1, CARD_2:57;
then reconsider P = (0,1) --> (x,y) as Function of (card {x,y}),{x,y} by A3, CARD_1:50, FUNCT_2:1;
A4: card (card {x,y}) = card {x,y} ;
A5: ( P . 0 = x & Fy . x = X ) by A1, A2, FUNCT_4:63;
A6: ( P . 1 = y & Fy . y = Y ) by A2, FUNCT_4:63;
A7: dom Fy = {x,y} by A2, FUNCT_4:62;
rng P = {x,y} by FUNCT_4:64;
then P is onto by FUNCT_2:def_3;
then P is one-to-one by A4, STIRL2_1:60;
then consider XFS being XFinSequence of such that
A8: dom XFS = card {x,y} and
A9: for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) and
A10: Card_Intersection (Fy,1) = Sum XFS by A7, Th44;
len XFS = 2 by A1, A8, CARD_2:57;
then A11: XFS = <%(XFS . 0),(XFS . 1)%> by AFINSQ_1:38;
A12: dom P = {0,1} by FUNCT_4:62;
then 1 in dom P by TARSKI:def_2;
then A13: (Fy * P) . 1 = Fy . (P . 1) by FUNCT_1:13;
0 in {0} by TARSKI:def_1;
then A14: ({x,y} --> 0) " {0} = {x,y} by FUNCOP_1:14;
( Fy . x = X & Fy . y = Y ) by A1, A2, FUNCT_4:63;
then A15: Intersection (Fy,({x,y} --> 0),0) = X /\ Y by A14, Th36;
0 in dom P by A12, TARSKI:def_2;
then A16: (Fy * P) . 0 = Fy . (P . 0) by FUNCT_1:13;
A17: dom XFS = 2 by A1, A8, CARD_2:57;
then 1 in dom XFS by CARD_1:50, TARSKI:def_2;
then A18: XFS . 1 = card Y by A9, A6, A13;
0 in dom XFS by A17, CARD_1:50, TARSKI:def_2;
then XFS . 0 = card X by A9, A5, A16;
then addnat "**" XFS = addnat . ((card X),(card Y)) by A11, A18, AFINSQ_2:38;
then A19: addnat "**" XFS = (card X) + (card Y) by BINOP_2:def_23;
( card {x,y} = 2 & dom Fy = {x,y} ) by A1, A2, CARD_2:57, FUNCT_4:62;
hence ( Card_Intersection (Fy,1) = (card X) + (card Y) & Card_Intersection (Fy,2) = card (X /\ Y) ) by A10, A19, A15, Th45, AFINSQ_2:51; ::_thesis: verum
end;
theorem Th48: :: CARD_FIN:48
for Fy being finite-yielding Function
for x being set st dom Fy is finite & x in dom Fy holds
Card_Intersection (Fy,1) = (Card_Intersection ((Fy | ((dom Fy) \ {x})),1)) + (card (Fy . x))
proof
let Fy be finite-yielding Function; ::_thesis: for x being set st dom Fy is finite & x in dom Fy holds
Card_Intersection (Fy,1) = (Card_Intersection ((Fy | ((dom Fy) \ {x})),1)) + (card (Fy . x))
let x be set ; ::_thesis: ( dom Fy is finite & x in dom Fy implies Card_Intersection (Fy,1) = (Card_Intersection ((Fy | ((dom Fy) \ {x})),1)) + (card (Fy . x)) )
assume that
A1: dom Fy is finite and
A2: x in dom Fy ; ::_thesis: Card_Intersection (Fy,1) = (Card_Intersection ((Fy | ((dom Fy) \ {x})),1)) + (card (Fy . x))
reconsider X = dom Fy as finite set by A1;
card X > 0 by A2;
then reconsider k = (card X) - 1 as Element of NAT by NAT_1:20;
set Xx = X \ {x};
A3: ( X \ {x} = {} implies card (X \ {x}) = {} ) ;
consider Px being Function of (card (X \ {x})),(X \ {x}) such that
A4: Px is one-to-one by Lm2;
not card (X \ {x}) in card (X \ {x}) ;
then consider P being Function of ((card (X \ {x})) \/ {(card (X \ {x}))}),((X \ {x}) \/ {x}) such that
A5: P | (card (X \ {x})) = Px and
A6: P . (card (X \ {x})) = x by A3, STIRL2_1:57;
not x in X \ {x} by ZFMISC_1:56;
then A7: P is one-to-one by A4, A3, A5, A6, STIRL2_1:58;
A8: card X = k + 1 ;
then A9: card (X \ {x}) = k by A2, STIRL2_1:55;
then card X = (card (X \ {x})) \/ {(card (X \ {x}))} by A8, AFINSQ_1:2;
then reconsider P = P as Function of (card X),X by A2, ZFMISC_1:116;
consider XFS being XFinSequence of such that
A10: dom XFS = card X and
A11: for z being set st z in dom XFS holds
XFS . z = card ((Fy * P) . z) and
A12: Card_Intersection (Fy,1) = Sum XFS by A7, Th44;
A13: P . k = x by A2, A6, A8, STIRL2_1:55;
X /\ (X \ {x}) = X \ {x} by XBOOLE_1:28;
then dom (Fy | (X \ {x})) = X \ {x} by RELAT_1:61;
then consider XFSx being XFinSequence of such that
A14: dom XFSx = card (X \ {x}) and
A15: for z being set st z in dom XFSx holds
XFSx . z = card (((Fy | (X \ {x})) * Px) . z) and
A16: Card_Intersection ((Fy | (X \ {x})),1) = Sum XFSx by A4, Th44;
k < k + 1 by NAT_1:13;
then A17: k c= k + 1 by NAT_1:39;
A18: for y being set st y in dom XFSx holds
XFS . y = XFSx . y
proof
A19: ( X \ {x} = X /\ (X \ {x}) & X /\ (X \ {x}) = dom (Fy | (X \ {x})) ) by RELAT_1:61, XBOOLE_1:28;
let y be set ; ::_thesis: ( y in dom XFSx implies XFS . y = XFSx . y )
assume A20: y in dom XFSx ; ::_thesis: XFS . y = XFSx . y
A21: XFS . y = card ((Fy * P) . y) by A14, A9, A10, A11, A17, A20;
A22: dom Px = k by A3, A9, FUNCT_2:def_1;
then Px . y in rng Px by A14, A9, A20, FUNCT_1:def_3;
then A23: (Fy | (X \ {x})) . (Px . y) = Fy . (Px . y) by A19, FUNCT_1:47;
dom P = k + 1 by CARD_1:27, FUNCT_2:def_1;
then A24: (Fy * P) . y = Fy . (P . y) by A14, A9, A17, A20, FUNCT_1:13;
Px . y = P . y by A14, A5, A9, A20, A22, FUNCT_1:47;
then (Fy * P) . y = ((Fy | (X \ {x})) * Px) . y by A14, A9, A20, A22, A24, A23, FUNCT_1:13;
hence XFS . y = XFSx . y by A15, A20, A21; ::_thesis: verum
end;
k < k + 1 by NAT_1:13;
then A25: k in card X by NAT_1:44;
then k in dom P by CARD_1:27, FUNCT_2:def_1;
then A26: (Fy * P) . k = Fy . (P . k) by FUNCT_1:13;
(dom XFS) /\ k = dom XFSx by A14, A9, A10, A17, XBOOLE_1:28;
then XFS | k = XFSx by A18, FUNCT_1:46;
then A27: (Sum XFSx) + (XFS . k) = Sum (XFS | (k + 1)) by A10, A25, AFINSQ_2:65;
XFS . k = card ((Fy * P) . k) by A10, A11, A25;
hence Card_Intersection (Fy,1) = (Card_Intersection ((Fy | ((dom Fy) \ {x})),1)) + (card (Fy . x)) by A16, A10, A12, A27, A26, A13, RELAT_1:69; ::_thesis: verum
end;
theorem Th49: :: CARD_FIN:49
for X9 being set
for F being Function holds
( dom (Intersect (F,((dom F) --> X9))) = dom F & ( for x being set st x in dom F holds
(Intersect (F,((dom F) --> X9))) . x = (F . x) /\ X9 ) )
proof
let X9 be set ; ::_thesis: for F being Function holds
( dom (Intersect (F,((dom F) --> X9))) = dom F & ( for x being set st x in dom F holds
(Intersect (F,((dom F) --> X9))) . x = (F . x) /\ X9 ) )
let F be Function; ::_thesis: ( dom (Intersect (F,((dom F) --> X9))) = dom F & ( for x being set st x in dom F holds
(Intersect (F,((dom F) --> X9))) . x = (F . x) /\ X9 ) )
dom ((dom F) --> X9) = dom F by FUNCOP_1:13;
then A1: (dom F) /\ (dom ((dom F) --> X9)) = dom F ;
hence dom F = dom (Intersect (F,((dom F) --> X9))) by YELLOW20:def_2; ::_thesis: for x being set st x in dom F holds
(Intersect (F,((dom F) --> X9))) . x = (F . x) /\ X9
let x be set ; ::_thesis: ( x in dom F implies (Intersect (F,((dom F) --> X9))) . x = (F . x) /\ X9 )
assume A2: x in dom F ; ::_thesis: (Intersect (F,((dom F) --> X9))) . x = (F . x) /\ X9
then (Intersect (F,((dom F) --> X9))) . x = (F . x) /\ (((dom F) --> X9) . x) by A1, YELLOW20:def_2;
hence (Intersect (F,((dom F) --> X9))) . x = (F . x) /\ X9 by A2, FUNCOP_1:7; ::_thesis: verum
end;
theorem Th50: :: CARD_FIN:50
for X9 being set
for F being Function holds (union (rng F)) /\ X9 = union (rng (Intersect (F,((dom F) --> X9))))
proof
let X9 be set ; ::_thesis: for F being Function holds (union (rng F)) /\ X9 = union (rng (Intersect (F,((dom F) --> X9))))
let F be Function; ::_thesis: (union (rng F)) /\ X9 = union (rng (Intersect (F,((dom F) --> X9))))
set I = Intersect (F,((dom F) --> X9));
thus (union (rng F)) /\ X9 c= union (rng (Intersect (F,((dom F) --> X9)))) :: according to XBOOLE_0:def_10 ::_thesis: union (rng (Intersect (F,((dom F) --> X9)))) c= (union (rng F)) /\ X9
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (union (rng F)) /\ X9 or x in union (rng (Intersect (F,((dom F) --> X9)))) )
assume A1: x in (union (rng F)) /\ X9 ; ::_thesis: x in union (rng (Intersect (F,((dom F) --> X9))))
A2: x in X9 by A1, XBOOLE_0:def_4;
x in union (rng F) by A1, XBOOLE_0:def_4;
then consider Fx being set such that
A3: x in Fx and
A4: Fx in rng F by TARSKI:def_4;
consider x1 being set such that
A5: x1 in dom F and
A6: F . x1 = Fx by A4, FUNCT_1:def_3;
x1 in dom (Intersect (F,((dom F) --> X9))) by A5, Th49;
then A7: (Intersect (F,((dom F) --> X9))) . x1 in rng (Intersect (F,((dom F) --> X9))) by FUNCT_1:def_3;
(Intersect (F,((dom F) --> X9))) . x1 = Fx /\ X9 by A5, A6, Th49;
then x in (Intersect (F,((dom F) --> X9))) . x1 by A3, A2, XBOOLE_0:def_4;
hence x in union (rng (Intersect (F,((dom F) --> X9)))) by A7, TARSKI:def_4; ::_thesis: verum
end;
thus union (rng (Intersect (F,((dom F) --> X9)))) c= (union (rng F)) /\ X9 ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union (rng (Intersect (F,((dom F) --> X9)))) or x in (union (rng F)) /\ X9 )
assume x in union (rng (Intersect (F,((dom F) --> X9)))) ; ::_thesis: x in (union (rng F)) /\ X9
then consider Ix being set such that
A8: x in Ix and
A9: Ix in rng (Intersect (F,((dom F) --> X9))) by TARSKI:def_4;
consider x1 being set such that
A10: x1 in dom (Intersect (F,((dom F) --> X9))) and
A11: (Intersect (F,((dom F) --> X9))) . x1 = Ix by A9, FUNCT_1:def_3;
A12: x1 in dom F by A10, Th49;
then A13: F . x1 in rng F by FUNCT_1:def_3;
A14: (Intersect (F,((dom F) --> X9))) . x1 = (F . x1) /\ X9 by A12, Th49;
then x in F . x1 by A8, A11, XBOOLE_0:def_4;
then A15: x in union (rng F) by A13, TARSKI:def_4;
x in X9 by A8, A11, A14, XBOOLE_0:def_4;
hence x in (union (rng F)) /\ X9 by A15, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
theorem Th51: :: CARD_FIN:51
for y, X9 being set
for F, Ch being Function holds (Intersection (F,Ch,y)) /\ X9 = Intersection ((Intersect (F,((dom F) --> X9))),Ch,y)
proof
let y, X9 be set ; ::_thesis: for F, Ch being Function holds (Intersection (F,Ch,y)) /\ X9 = Intersection ((Intersect (F,((dom F) --> X9))),Ch,y)
let F, Ch be Function; ::_thesis: (Intersection (F,Ch,y)) /\ X9 = Intersection ((Intersect (F,((dom F) --> X9))),Ch,y)
set I = Intersect (F,((dom F) --> X9));
set Int1 = Intersection (F,Ch,y);
set Int2 = Intersection ((Intersect (F,((dom F) --> X9))),Ch,y);
thus (Intersection (F,Ch,y)) /\ X9 c= Intersection ((Intersect (F,((dom F) --> X9))),Ch,y) :: according to XBOOLE_0:def_10 ::_thesis: Intersection ((Intersect (F,((dom F) --> X9))),Ch,y) c= (Intersection (F,Ch,y)) /\ X9
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (Intersection (F,Ch,y)) /\ X9 or x in Intersection ((Intersect (F,((dom F) --> X9))),Ch,y) )
assume A1: x in (Intersection (F,Ch,y)) /\ X9 ; ::_thesis: x in Intersection ((Intersect (F,((dom F) --> X9))),Ch,y)
A2: for z being set st z in dom Ch & Ch . z = y holds
x in (Intersect (F,((dom F) --> X9))) . z
proof
A3: x in Intersection (F,Ch,y) by A1, XBOOLE_0:def_4;
let z be set ; ::_thesis: ( z in dom Ch & Ch . z = y implies x in (Intersect (F,((dom F) --> X9))) . z )
assume ( z in dom Ch & Ch . z = y ) ; ::_thesis: x in (Intersect (F,((dom F) --> X9))) . z
then A4: x in F . z by A3, Def2;
then A5: z in dom F by FUNCT_1:def_2;
x in X9 by A1, XBOOLE_0:def_4;
then x in (F . z) /\ X9 by A4, XBOOLE_0:def_4;
hence x in (Intersect (F,((dom F) --> X9))) . z by A5, Th49; ::_thesis: verum
end;
x in X9 by A1, XBOOLE_0:def_4;
then x in (union (rng F)) /\ X9 by A1, XBOOLE_0:def_4;
then x in union (rng (Intersect (F,((dom F) --> X9)))) by Th50;
hence x in Intersection ((Intersect (F,((dom F) --> X9))),Ch,y) by A2, Def2; ::_thesis: verum
end;
thus Intersection ((Intersect (F,((dom F) --> X9))),Ch,y) c= (Intersection (F,Ch,y)) /\ X9 ::_thesis: verum
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in Intersection ((Intersect (F,((dom F) --> X9))),Ch,y) or x in (Intersection (F,Ch,y)) /\ X9 )
assume A6: x in Intersection ((Intersect (F,((dom F) --> X9))),Ch,y) ; ::_thesis: x in (Intersection (F,Ch,y)) /\ X9
x in union (rng (Intersect (F,((dom F) --> X9)))) by A6;
then A7: x in (union (rng F)) /\ X9 by Th50;
then A8: x in X9 by XBOOLE_0:def_4;
A9: for z being set st z in dom Ch & Ch . z = y holds
x in F . z
proof
A10: dom (Intersect (F,((dom F) --> X9))) = dom F by Th49;
let z be set ; ::_thesis: ( z in dom Ch & Ch . z = y implies x in F . z )
assume ( z in dom Ch & Ch . z = y ) ; ::_thesis: x in F . z
then A11: x in (Intersect (F,((dom F) --> X9))) . z by A6, Def2;
then z in dom (Intersect (F,((dom F) --> X9))) by FUNCT_1:def_2;
then x in (F . z) /\ X9 by A11, A10, Th49;
hence x in F . z by XBOOLE_0:def_4; ::_thesis: verum
end;
x in union (rng F) by A7, XBOOLE_0:def_4;
then x in Intersection (F,Ch,y) by A9, Def2;
hence x in (Intersection (F,Ch,y)) /\ X9 by A8, XBOOLE_0:def_4; ::_thesis: verum
end;
end;
theorem Th52: :: CARD_FIN:52
for F, G being XFinSequence st F is one-to-one & G is one-to-one & rng F misses rng G holds
F ^ G is one-to-one
proof
let F, G be XFinSequence; ::_thesis: ( F is one-to-one & G is one-to-one & rng F misses rng G implies F ^ G is one-to-one )
assume that
A1: F is one-to-one and
A2: G is one-to-one and
A3: rng F misses rng G ; ::_thesis: F ^ G is one-to-one
len F, rng F are_equipotent by A1, WELLORD2:def_4;
then A4: card (len F) = card (rng F) by CARD_1:5;
len G, rng G are_equipotent by A2, WELLORD2:def_4;
then A5: card (len G) = card (rng G) by CARD_1:5;
reconsider FG = F ^ G as Function of (dom (F ^ G)),(rng (F ^ G)) by FUNCT_2:1;
A6: dom (F ^ G) = (len F) + (len G) by AFINSQ_1:def_3;
A7: FG is onto by FUNCT_2:def_3;
card ((rng F) \/ (rng G)) = (card (rng F)) + (card (rng G)) by A3, CARD_2:40;
then card (dom (F ^ G)) = card (rng (F ^ G)) by A4, A5, A6, AFINSQ_1:26;
hence F ^ G is one-to-one by A7, STIRL2_1:60; ::_thesis: verum
end;
theorem Th53: :: CARD_FIN:53
for k being Nat
for Fy being finite-yielding Function
for X being finite set
for x being set
for n being Nat st dom Fy = X & x in dom Fy & k > 0 holds
Card_Intersection (Fy,(k + 1)) = (Card_Intersection ((Fy | ((dom Fy) \ {x})),(k + 1))) + (Card_Intersection ((Intersect ((Fy | ((dom Fy) \ {x})),(((dom Fy) \ {x}) --> (Fy . x)))),k))
proof
let k be Nat; ::_thesis: for Fy being finite-yielding Function
for X being finite set
for x being set
for n being Nat st dom Fy = X & x in dom Fy & k > 0 holds
Card_Intersection (Fy,(k + 1)) = (Card_Intersection ((Fy | ((dom Fy) \ {x})),(k + 1))) + (Card_Intersection ((Intersect ((Fy | ((dom Fy) \ {x})),(((dom Fy) \ {x}) --> (Fy . x)))),k))
let Fy be finite-yielding Function; ::_thesis: for X being finite set
for x being set
for n being Nat st dom Fy = X & x in dom Fy & k > 0 holds
Card_Intersection (Fy,(k + 1)) = (Card_Intersection ((Fy | ((dom Fy) \ {x})),(k + 1))) + (Card_Intersection ((Intersect ((Fy | ((dom Fy) \ {x})),(((dom Fy) \ {x}) --> (Fy . x)))),k))
let X be finite set ; ::_thesis: for x being set
for n being Nat st dom Fy = X & x in dom Fy & k > 0 holds
Card_Intersection (Fy,(k + 1)) = (Card_Intersection ((Fy | ((dom Fy) \ {x})),(k + 1))) + (Card_Intersection ((Intersect ((Fy | ((dom Fy) \ {x})),(((dom Fy) \ {x}) --> (Fy . x)))),k))
let x be set ; ::_thesis: for n being Nat st dom Fy = X & x in dom Fy & k > 0 holds
Card_Intersection (Fy,(k + 1)) = (Card_Intersection ((Fy | ((dom Fy) \ {x})),(k + 1))) + (Card_Intersection ((Intersect ((Fy | ((dom Fy) \ {x})),(((dom Fy) \ {x}) --> (Fy . x)))),k))
let n be Nat; ::_thesis: ( dom Fy = X & x in dom Fy & k > 0 implies Card_Intersection (Fy,(k + 1)) = (Card_Intersection ((Fy | ((dom Fy) \ {x})),(k + 1))) + (Card_Intersection ((Intersect ((Fy | ((dom Fy) \ {x})),(((dom Fy) \ {x}) --> (Fy . x)))),k)) )
assume that
A1: dom Fy = X and
A2: x in dom Fy and
A3: k > 0 ; ::_thesis: Card_Intersection (Fy,(k + 1)) = (Card_Intersection ((Fy | ((dom Fy) \ {x})),(k + 1))) + (Card_Intersection ((Intersect ((Fy | ((dom Fy) \ {x})),(((dom Fy) \ {x}) --> (Fy . x)))),k))
set Xx = X \ {x};
A4: (X \ {x}) \/ {x} = X by A1, A2, ZFMISC_1:116;
set I = Intersect ((Fy | ((dom Fy) \ {x})),(((dom Fy) \ {x}) --> (Fy . x)));
set X1 = { f where f is Function of ((X \ {x}) \/ {x}),{1,0} : ( card (f " {1}) = k + 1 & f . x = 1 ) } ;
set X0 = { f where f is Function of ((X \ {x}) \/ {x}),{1,0} : ( card (f " {1}) = k + 1 & f . x = 0 ) } ;
{ f where f is Function of ((X \ {x}) \/ {x}),{1,0} : ( card (f " {1}) = k + 1 & f . x = 0 ) } \/ { f where f is Function of ((X \ {x}) \/ {x}),{1,0} : ( card (f " {1}) = k + 1 & f . x = 1 ) } = Choose (((X \ {x}) \/ {x}),(k + 1),1,0) by Lm1;
then reconsider X0 = { f where f is Function of ((X \ {x}) \/ {x}),{1,0} : ( card (f " {1}) = k + 1 & f . x = 0 ) } , X1 = { f where f is Function of ((X \ {x}) \/ {x}),{1,0} : ( card (f " {1}) = k + 1 & f . x = 1 ) } as finite set by FINSET_1:1, XBOOLE_1:7;
consider P1 being Function of (card X1),X1 such that
A5: P1 is one-to-one by Lm2;
not x in X \ {x} by ZFMISC_1:56;
then A6: card (Choose ((X \ {x}),k,1,0)) = card X1 by Th13;
defpred S1[ set , set ] means ex f being Function st
( f = P1 . $1 & f in X1 & $2 = f | (X \ {x}) );
A7: for x1 being set st x1 in card X1 holds
ex P1x1 being set st
( P1x1 in Choose ((X \ {x}),k,1,0) & S1[x1,P1x1] )
proof
not x in X \ {x} by ZFMISC_1:56;
then A8: ((X \ {x}) \/ {x}) \ {x} = X \ {x} by ZFMISC_1:117;
let x1 be set ; ::_thesis: ( x1 in card X1 implies ex P1x1 being set st
( P1x1 in Choose ((X \ {x}),k,1,0) & S1[x1,P1x1] ) )
assume x1 in card X1 ; ::_thesis: ex P1x1 being set st
( P1x1 in Choose ((X \ {x}),k,1,0) & S1[x1,P1x1] )
then x1 in dom P1 by CARD_1:27, FUNCT_2:def_1;
then A9: P1 . x1 in rng P1 by FUNCT_1:def_3;
then P1 . x1 in X1 ;
then consider P1x1 being Function of ((X \ {x}) \/ {x}),{1,0} such that
A10: P1 . x1 = P1x1 and
A11: card (P1x1 " {1}) = k + 1 and
A12: P1x1 . x = 1 ;
A13: dom P1x1 = (X \ {x}) \/ {x} by FUNCT_2:def_1;
A14: rng (P1x1 | (X \ {x})) c= {1,0} ;
((X \ {x}) \/ {x}) /\ (X \ {x}) = X \ {x} by XBOOLE_1:7, XBOOLE_1:28;
then dom (P1x1 | (X \ {x})) = X \ {x} by A13, RELAT_1:61;
then reconsider Px = P1x1 | (X \ {x}) as Function of (X \ {x}),{1,0} by A14, FUNCT_2:2;
A15: not x in Px " {1} by ZFMISC_1:56;
( x in {x} & dom P1x1 = (X \ {x}) \/ {x} ) by FUNCT_2:def_1, TARSKI:def_1;
then x in dom P1x1 by XBOOLE_0:def_3;
then P1x1 " {1} = (Px " {1}) \/ {x} by A12, A13, A8, AFINSQ_2:66;
then k + 1 = (card (Px " {1})) + 1 by A11, A15, CARD_2:41;
then Px in Choose ((X \ {x}),k,1,0) by Def1;
hence ex P1x1 being set st
( P1x1 in Choose ((X \ {x}),k,1,0) & S1[x1,P1x1] ) by A9, A10; ::_thesis: verum
end;
consider P1x being Function of (card X1),(Choose ((X \ {x}),k,1,0)) such that
A16: for x1 being set st x1 in card X1 holds
S1[x1,P1x . x1] from FUNCT_2:sch_1(A7);
for x1, x2 being set st x1 in dom P1x & x2 in dom P1x & P1x . x1 = P1x . x2 holds
x1 = x2
proof
let x1, x2 be set ; ::_thesis: ( x1 in dom P1x & x2 in dom P1x & P1x . x1 = P1x . x2 implies x1 = x2 )
assume that
A17: x1 in dom P1x and
A18: x2 in dom P1x and
A19: P1x . x1 = P1x . x2 ; ::_thesis: x1 = x2
consider f2 being Function such that
A20: f2 = P1 . x2 and
A21: f2 in X1 and
A22: P1x . x2 = f2 | (X \ {x}) by A16, A18;
consider f1 being Function such that
A23: f1 = P1 . x1 and
A24: f1 in X1 and
A25: P1x . x1 = f1 | (X \ {x}) by A16, A17;
A26: ex F being Function of ((X \ {x}) \/ {x}),{1,0} st
( f1 = F & card (F " {1}) = k + 1 & F . x = 1 ) by A24;
then A27: dom f1 = (X \ {x}) \/ {x} by FUNCT_2:def_1;
A28: ex F being Function of ((X \ {x}) \/ {x}),{1,0} st
( f2 = F & card (F " {1}) = k + 1 & F . x = 1 ) by A21;
then A29: dom f2 = (X \ {x}) \/ {x} by FUNCT_2:def_1;
for z being set st z in dom f1 holds
f1 . z = f2 . z
proof
let z be set ; ::_thesis: ( z in dom f1 implies f1 . z = f2 . z )
assume A30: z in dom f1 ; ::_thesis: f1 . z = f2 . z
now__::_thesis:_f1_._z_=_f2_._z
percases ( z in X \ {x} or z in {x} ) by A27, A30, XBOOLE_0:def_3;
supposeA31: z in X \ {x} ; ::_thesis: f1 . z = f2 . z
then z in (dom f1) /\ (X \ {x}) by A30, XBOOLE_0:def_4;
then A32: (f1 | (X \ {x})) . z = f1 . z by FUNCT_1:48;
z in (dom f2) /\ (X \ {x}) by A27, A29, A30, A31, XBOOLE_0:def_4;
hence f1 . z = f2 . z by A19, A25, A22, A32, FUNCT_1:48; ::_thesis: verum
end;
suppose z in {x} ; ::_thesis: f1 . z = f2 . z
then z = x by TARSKI:def_1;
hence f1 . z = f2 . z by A26, A28; ::_thesis: verum
end;
end;
end;
hence f1 . z = f2 . z ; ::_thesis: verum
end;
then A33: f1 = f2 by A27, A29, FUNCT_1:2;
( X1 = {} implies card X1 = {} ) ;
then dom P1 = card X1 by FUNCT_2:def_1;
hence x1 = x2 by A5, A17, A18, A23, A20, A33, FUNCT_1:def_4; ::_thesis: verum
end;
then A34: P1x is one-to-one by FUNCT_1:def_4;
(X \ {x}) /\ X = X \ {x} by XBOOLE_1:28;
then A35: dom (Fy | ((dom Fy) \ {x})) = X \ {x} by A1, RELAT_1:61;
then dom (Intersect ((Fy | ((dom Fy) \ {x})),(((dom Fy) \ {x}) --> (Fy . x)))) = X \ {x} by A1, Th49;
then consider XFS1 being XFinSequence of such that
A36: dom XFS1 = dom P1x and
A37: for z being set
for f being Function st z in dom XFS1 & f = P1x . z holds
XFS1 . z = card (Intersection ((Intersect ((Fy | ((dom Fy) \ {x})),(((dom Fy) \ {x}) --> (Fy . x)))),f,1)) and
A38: Card_Intersection ((Intersect ((Fy | ((dom Fy) \ {x})),(((dom Fy) \ {x}) --> (Fy . x)))),k) = Sum XFS1 by A6, A34, Def4;
A39: addnat "**" XFS1 = Card_Intersection ((Intersect ((Fy | ((dom Fy) \ {x})),(((dom Fy) \ {x}) --> (Fy . x)))),k) by A38, AFINSQ_2:51;
not x in X \ {x} by ZFMISC_1:56;
then A40: card (Choose ((X \ {x}),(k + 1),1,0)) = card X0 by Th14;
set Ch = Choose (X,(k + 1),1,0);
consider P0 being Function of (card X0),X0 such that
A41: P0 is one-to-one by Lm2;
A42: ( X1 = {} implies card X1 = {} ) ;
then A43: dom P1 = card X1 by FUNCT_2:def_1;
A44: ( X0 = {} implies card X0 = {} ) ;
then dom P0 = card X0 by FUNCT_2:def_1;
then reconsider XP0 = P0, XP1 = P1 as XFinSequence by A43, AFINSQ_1:5;
A45: card X0 = len XP0 by A44, FUNCT_2:def_1;
defpred S2[ set , set ] means ex f being Function st
( f = P0 . $1 & f in X0 & $2 = f | (X \ {x}) );
A46: for x0 being set st x0 in card X0 holds
ex P0x0 being set st
( P0x0 in Choose ((X \ {x}),(k + 1),1,0) & S2[x0,P0x0] )
proof
let x0 be set ; ::_thesis: ( x0 in card X0 implies ex P0x0 being set st
( P0x0 in Choose ((X \ {x}),(k + 1),1,0) & S2[x0,P0x0] ) )
assume x0 in card X0 ; ::_thesis: ex P0x0 being set st
( P0x0 in Choose ((X \ {x}),(k + 1),1,0) & S2[x0,P0x0] )
then x0 in dom P0 by CARD_1:27, FUNCT_2:def_1;
then A47: P0 . x0 in rng P0 by FUNCT_1:def_3;
then P0 . x0 in X0 ;
then consider P0x0 being Function of ((X \ {x}) \/ {x}),{1,0} such that
A48: P0 . x0 = P0x0 and
A49: card (P0x0 " {1}) = k + 1 and
A50: P0x0 . x = 0 ;
A51: dom P0x0 = (X \ {x}) \/ {x} by FUNCT_2:def_1;
A52: rng (P0x0 | (X \ {x})) c= {1,0} ;
((X \ {x}) \/ {x}) /\ (X \ {x}) = X \ {x} by XBOOLE_1:7, XBOOLE_1:28;
then dom (P0x0 | (X \ {x})) = X \ {x} by A51, RELAT_1:61;
then reconsider Px = P0x0 | (X \ {x}) as Function of (X \ {x}),{1,0} by A52, FUNCT_2:2;
not x in X \ {x} by ZFMISC_1:56;
then ((X \ {x}) \/ {x}) \ {x} = X \ {x} by ZFMISC_1:117;
then P0x0 " {1} = Px " {1} by A50, A51, AFINSQ_2:67;
then Px in Choose ((X \ {x}),(k + 1),1,0) by A49, Def1;
hence ex P0x0 being set st
( P0x0 in Choose ((X \ {x}),(k + 1),1,0) & S2[x0,P0x0] ) by A47, A48; ::_thesis: verum
end;
consider P0x being Function of (card X0),(Choose ((X \ {x}),(k + 1),1,0)) such that
A53: for x1 being set st x1 in card X0 holds
S2[x1,P0x . x1] from FUNCT_2:sch_1(A46);
(rng P0) \/ (rng P1) c= X0 \/ X1 by XBOOLE_1:13;
then rng (XP0 ^ XP1) c= X0 \/ X1 by AFINSQ_1:26;
then A54: rng (XP0 ^ XP1) c= Choose (X,(k + 1),1,0) by A4, Lm1;
A55: card X1 = len XP1 by A42, FUNCT_2:def_1;
for x1, x2 being set st x1 in dom P0x & x2 in dom P0x & P0x . x1 = P0x . x2 holds
x1 = x2
proof
let x1, x2 be set ; ::_thesis: ( x1 in dom P0x & x2 in dom P0x & P0x . x1 = P0x . x2 implies x1 = x2 )
assume that
A56: x1 in dom P0x and
A57: x2 in dom P0x and
A58: P0x . x1 = P0x . x2 ; ::_thesis: x1 = x2
consider f2 being Function such that
A59: f2 = P0 . x2 and
A60: f2 in X0 and
A61: P0x . x2 = f2 | (X \ {x}) by A53, A57;
consider f1 being Function such that
A62: f1 = P0 . x1 and
A63: f1 in X0 and
A64: P0x . x1 = f1 | (X \ {x}) by A53, A56;
A65: ex F being Function of ((X \ {x}) \/ {x}),{1,0} st
( f1 = F & card (F " {1}) = k + 1 & F . x = 0 ) by A63;
then A66: dom f1 = (X \ {x}) \/ {x} by FUNCT_2:def_1;
A67: ex F being Function of ((X \ {x}) \/ {x}),{1,0} st
( f2 = F & card (F " {1}) = k + 1 & F . x = 0 ) by A60;
then A68: dom f2 = (X \ {x}) \/ {x} by FUNCT_2:def_1;
for z being set st z in dom f1 holds
f1 . z = f2 . z
proof
let z be set ; ::_thesis: ( z in dom f1 implies f1 . z = f2 . z )
assume A69: z in dom f1 ; ::_thesis: f1 . z = f2 . z
now__::_thesis:_f1_._z_=_f2_._z
percases ( z in X \ {x} or z in {x} ) by A66, A69, XBOOLE_0:def_3;
supposeA70: z in X \ {x} ; ::_thesis: f1 . z = f2 . z
then z in (dom f1) /\ (X \ {x}) by A69, XBOOLE_0:def_4;
then A71: (f1 | (X \ {x})) . z = f1 . z by FUNCT_1:48;
z in (dom f2) /\ (X \ {x}) by A66, A68, A69, A70, XBOOLE_0:def_4;
hence f1 . z = f2 . z by A58, A64, A61, A71, FUNCT_1:48; ::_thesis: verum
end;
suppose z in {x} ; ::_thesis: f1 . z = f2 . z
then z = x by TARSKI:def_1;
hence f1 . z = f2 . z by A65, A67; ::_thesis: verum
end;
end;
end;
hence f1 . z = f2 . z ; ::_thesis: verum
end;
then A72: f1 = f2 by A66, A68, FUNCT_1:2;
( X0 = {} implies card X0 = {} ) ;
then dom P0 = card X0 by FUNCT_2:def_1;
hence x1 = x2 by A41, A56, A57, A62, A59, A72, FUNCT_1:def_4; ::_thesis: verum
end;
then P0x is one-to-one by FUNCT_1:def_4;
then consider XFS0 being XFinSequence of such that
A73: dom XFS0 = dom P0x and
A74: for z being set
for f being Function st z in dom XFS0 & f = P0x . z holds
XFS0 . z = card (Intersection ((Fy | ((dom Fy) \ {x})),f,1)) and
A75: Card_Intersection ((Fy | ((dom Fy) \ {x})),(k + 1)) = Sum XFS0 by A40, A35, Def4;
( Choose ((X \ {x}),(k + 1),1,0) = {} implies card (Choose ((X \ {x}),(k + 1),1,0)) = {} ) ;
then A76: dom P0x = card X0 by A40, FUNCT_2:def_1;
not x in X \ {x} by ZFMISC_1:56;
then (card X0) + (card X1) = card (Choose (X,(k + 1),1,0)) by A40, A6, A4, Th15;
then dom (XP0 ^ XP1) = card (Choose (X,(k + 1),1,0)) by A45, A55, AFINSQ_1:def_3;
then reconsider XP01 = XP0 ^ XP1 as Function of (card (Choose (X,(k + 1),1,0))),(Choose (X,(k + 1),1,0)) by A54, FUNCT_2:2;
rng P0 misses rng P1 by Lm1, XBOOLE_1:64;
then XP01 is one-to-one by A41, A5, Th52;
then consider XFS being XFinSequence of such that
A77: dom XFS = dom XP01 and
A78: for z being set
for f being Function st z in dom XFS & f = XP01 . z holds
XFS . z = card (Intersection (Fy,f,1)) and
A79: Card_Intersection (Fy,(k + 1)) = Sum XFS by A1, Def4;
A80: addnat "**" XFS = Card_Intersection (Fy,(k + 1)) by A79, AFINSQ_2:51;
( Choose ((X \ {x}),k,1,0) = {} implies card (Choose ((X \ {x}),k,1,0)) = {} ) ;
then A81: dom P1x = card X1 by A6, FUNCT_2:def_1;
A82: for n being Nat st n in dom XFS0 holds
XFS . n = XFS0 . n
proof
let n be Nat; ::_thesis: ( n in dom XFS0 implies XFS . n = XFS0 . n )
assume A83: n in dom XFS0 ; ::_thesis: XFS . n = XFS0 . n
consider fx being Function such that
A84: fx = P0 . n and
A85: fx in X0 and
A86: P0x . n = fx | (X \ {x}) by A53, A73, A83;
A87: XFS0 . n = card (Intersection ((Fy | (X \ {x})),(fx | (X \ {x})),1)) by A1, A74, A83, A86;
A88: ex fx9 being Function of ((X \ {x}) \/ {x}),{1,0} st
( fx = fx9 & card (fx9 " {1}) = k + 1 & fx9 . x = 0 ) by A85;
then consider x1 being set such that
A89: x1 in fx " {1} by CARD_1:27, XBOOLE_0:def_1;
fx . x1 in {1} by A89, FUNCT_1:def_7;
then A90: fx . x1 = 1 by TARSKI:def_1;
x1 in dom fx by A89, FUNCT_1:def_7;
then A91: 1 in rng fx by A90, FUNCT_1:def_3;
A92: (X \ {x}) \/ {x} = X by A1, A2, ZFMISC_1:116;
A93: (dom XFS0) + 0 <= (dom XFS0) + (dom XFS1) by XREAL_1:7;
dom fx = (X \ {x}) \/ {x} by A88, FUNCT_2:def_1;
then A94: fx " {1} = (fx | (X \ {x})) " {1} by A88, A92, AFINSQ_2:67;
n < dom XFS0 by A83, NAT_1:44;
then n < (dom XFS0) + (dom XFS1) by A93, XXREAL_0:2;
then n < dom XFS by A73, A36, A45, A55, A77, A76, A81, AFINSQ_1:def_3;
then A95: n in dom XFS by NAT_1:44;
XP01 . n = XP0 . n by A73, A45, A83, AFINSQ_1:def_3;
then A96: XFS . n = card (Intersection (Fy,fx,1)) by A78, A84, A95;
( (fx | (X \ {x})) " {1} c= dom (fx | (X \ {x})) & dom (fx | (X \ {x})) c= X \ {x} ) by RELAT_1:58, RELAT_1:132;
then Intersection ((Fy | (X \ {x})),fx,1) = Intersection (Fy,fx,1) by A94, A91, Th30, XBOOLE_1:1;
hence XFS . n = XFS0 . n by A94, A96, A87, Th28; ::_thesis: verum
end;
( X1 = {} implies card X1 = {} ) ;
then A97: dom P1 = card X1 by FUNCT_2:def_1;
A98: for n being Nat st n in dom XFS1 holds
XFS . ((len XFS0) + n) = XFS1 . n
proof
A99: (X \ {x}) \/ {x} = X by A1, A2, ZFMISC_1:116;
let n be Nat; ::_thesis: ( n in dom XFS1 implies XFS . ((len XFS0) + n) = XFS1 . n )
assume A100: n in dom XFS1 ; ::_thesis: XFS . ((len XFS0) + n) = XFS1 . n
consider fx being Function such that
A101: fx = P1 . n and
A102: fx in X1 and
A103: P1x . n = fx | (X \ {x}) by A16, A36, A100;
consider fx9 being Function of ((X \ {x}) \/ {x}),{1,0} such that
A104: fx = fx9 and
A105: card (fx9 " {1}) = k + 1 and
A106: fx9 . x = 1 by A102;
A107: Intersection ((Intersect ((Fy | (X \ {x})),((X \ {x}) --> (Fy . x)))),(fx | (X \ {x})),1) = (Intersection ((Fy | (X \ {x})),(fx | (X \ {x})),1)) /\ (Fy . x) by A1, A35, Th51;
A108: dom fx9 = (X \ {x}) \/ {x} by FUNCT_2:def_1;
then A109: dom fx = X by A1, A2, A104, ZFMISC_1:116;
A110: ( 1 in rng (fx | (X \ {x})) & (fx | (X \ {x})) " {1} c= X \ {x} )
proof
A111: ( (fx | (X \ {x})) " {1} c= dom (fx | (X \ {x})) & dom (fx | (X \ {x})) = (dom fx) /\ (X \ {x}) ) by RELAT_1:61, RELAT_1:132;
reconsider fx1 = (fx | (X \ {x})) " {1} as finite set ;
not x in X \ {x} by ZFMISC_1:56;
then not x in (dom fx) /\ (X \ {x}) by XBOOLE_0:def_4;
then not x in dom (fx | (X \ {x})) by RELAT_1:61;
then A112: not x in fx1 by FUNCT_1:def_7;
{x} \/ fx1 = fx " {1} by A1, A2, A104, A106, A109, AFINSQ_2:66;
then (card fx1) + 1 = k + 1 by A104, A105, A112, CARD_2:41;
then consider y being set such that
A113: y in fx1 by A3, CARD_1:27, XBOOLE_0:def_1;
y in dom (fx | (X \ {x})) by A113, FUNCT_1:def_7;
then A114: (fx | (X \ {x})) . y in rng (fx | (X \ {x})) by FUNCT_1:def_3;
( (dom fx) /\ (X \ {x}) c= X \ {x} & (fx | (X \ {x})) . y in {1} ) by A113, FUNCT_1:def_7, XBOOLE_1:17;
hence ( 1 in rng (fx | (X \ {x})) & (fx | (X \ {x})) " {1} c= X \ {x} ) by A111, A114, TARSKI:def_1, XBOOLE_1:1; ::_thesis: verum
end;
n < dom XFS1 by A100, NAT_1:44;
then (dom XFS0) + n < (dom XFS0) + (dom XFS1) by XREAL_1:8;
then (dom XFS0) + n < dom XFS by A73, A36, A45, A55, A77, A76, A81, AFINSQ_1:def_3;
then A115: (dom XFS0) + n in dom XFS by NAT_1:44;
XP01 . (n + (len XP0)) = fx by A36, A97, A100, A101, AFINSQ_1:def_3;
then A116: XFS . ((dom XFS0) + n) = card (Intersection (Fy,fx,1)) by A73, A45, A78, A76, A115;
fx . x in {1} by A104, A106, TARSKI:def_1;
then A117: x in fx " {1} by A1, A2, A104, A108, A99, FUNCT_1:def_7;
XFS1 . n = card (Intersection ((Intersect ((Fy | (X \ {x})),((X \ {x}) --> (Fy . x)))),(fx | (X \ {x})),1)) by A1, A37, A100, A103;
then XFS1 . n = card ((Intersection (Fy,(fx | (X \ {x})),1)) /\ (Fy . x)) by A110, A107, Th30;
hence XFS . ((len XFS0) + n) = XFS1 . n by A117, A109, A116, Th32; ::_thesis: verum
end;
dom XFS = (len XFS0) + (len XFS1) by A73, A36, A45, A55, A77, A76, A81, AFINSQ_1:def_3;
then XFS = XFS0 ^ XFS1 by A82, A98, AFINSQ_1:def_3;
then A118: addnat "**" XFS = addnat . ((addnat "**" XFS0),(addnat "**" XFS1)) by AFINSQ_2:42;
addnat "**" XFS0 = Card_Intersection ((Fy | ((dom Fy) \ {x})),(k + 1)) by A75, AFINSQ_2:51;
hence Card_Intersection (Fy,(k + 1)) = (Card_Intersection ((Fy | ((dom Fy) \ {x})),(k + 1))) + (Card_Intersection ((Intersect ((Fy | ((dom Fy) \ {x})),(((dom Fy) \ {x}) --> (Fy . x)))),k)) by A118, A39, A80, BINOP_2:def_23; ::_thesis: verum
end;
theorem Th54: :: CARD_FIN:54
for x being set
for F being Function st x in dom F holds
union (rng F) = (union (rng (F | ((dom F) \ {x})))) \/ (F . x)
proof
let x be set ; ::_thesis: for F being Function st x in dom F holds
union (rng F) = (union (rng (F | ((dom F) \ {x})))) \/ (F . x)
let F be Function; ::_thesis: ( x in dom F implies union (rng F) = (union (rng (F | ((dom F) \ {x})))) \/ (F . x) )
set d = (dom F) \ {x};
set Fd = F | ((dom F) \ {x});
A1: F | (dom F) = F ;
assume A2: x in dom F ; ::_thesis: union (rng F) = (union (rng (F | ((dom F) \ {x})))) \/ (F . x)
then ((dom F) \ {x}) \/ {x} = dom F by ZFMISC_1:116;
then F = (F | ((dom F) \ {x})) \/ (F | {x}) by A1, RELAT_1:78;
then A3: rng F = (rng (F | ((dom F) \ {x}))) \/ (rng (F | {x})) by RELAT_1:12;
Im (F,x) = {(F . x)} by A2, FUNCT_1:59;
then rng (F | {x}) = {(F . x)} by RELAT_1:115;
then union (rng F) = (union (rng (F | ((dom F) \ {x})))) \/ (union {(F . x)}) by A3, ZFMISC_1:78;
hence union (rng F) = (union (rng (F | ((dom F) \ {x})))) \/ (F . x) by ZFMISC_1:25; ::_thesis: verum
end;
theorem Th55: :: CARD_FIN:55
for Fy being finite-yielding Function
for X being finite set ex XFS being XFinSequence of st
( dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) )
proof
let Fy be finite-yielding Function; ::_thesis: for X being finite set ex XFS being XFinSequence of st
( dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) )
let X be finite set ; ::_thesis: ex XFS being XFinSequence of st
( dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) )
defpred S1[ set , set ] means for n being Nat st n = $1 holds
$2 = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1)));
A1: for k being Nat st k in card X holds
ex x being Element of INT st S1[k,x]
proof
let k be Nat; ::_thesis: ( k in card X implies ex x being Element of INT st S1[k,x] )
assume k in card X ; ::_thesis: ex x being Element of INT st S1[k,x]
reconsider C = ((- 1) |^ k) * (Card_Intersection (Fy,(k + 1))) as Element of INT ;
take C ; ::_thesis: S1[k,C]
thus S1[k,C] ; ::_thesis: verum
end;
consider XFS being XFinSequence of such that
A2: ( dom XFS = card X & ( for k being Nat st k in card X holds
S1[k,XFS . k] ) ) from STIRL2_1:sch_5(A1);
take XFS ; ::_thesis: ( dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) )
thus ( dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) ) by A2; ::_thesis: verum
end;
theorem Th56: :: CARD_FIN:56
for Fy being finite-yielding Function
for X being finite set st dom Fy = X holds
for XFS being XFinSequence of st dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) holds
card (union (rng Fy)) = Sum XFS
proof
defpred S1[ Nat] means for Fy being finite-yielding Function
for X being finite set st dom Fy = X & card X = $1 holds
for XFS being XFinSequence of st dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) holds
card (union (rng Fy)) = Sum XFS;
A1: for k being Nat st S1[k] holds
S1[k + 1]
proof
let k be Nat; ::_thesis: ( S1[k] implies S1[k + 1] )
assume A2: S1[k] ; ::_thesis: S1[k + 1]
let Fy be finite-yielding Function; ::_thesis: for X being finite set st dom Fy = X & card X = k + 1 holds
for XFS being XFinSequence of st dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) holds
card (union (rng Fy)) = Sum XFS
let X be finite set ; ::_thesis: ( dom Fy = X & card X = k + 1 implies for XFS being XFinSequence of st dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) holds
card (union (rng Fy)) = Sum XFS )
assume that
A3: dom Fy = X and
A4: card X = k + 1 ; ::_thesis: for XFS being XFinSequence of st dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) holds
card (union (rng Fy)) = Sum XFS
( rng Fy is finite & ( for x being set st x in rng Fy holds
x is finite ) ) by A3, FINSET_1:8;
then reconsider urngFy = union (rng Fy) as finite set ;
let XFS be XFinSequence of ; ::_thesis: ( dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) implies card (union (rng Fy)) = Sum XFS )
assume that
A5: dom XFS = card X and
A6: for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ; ::_thesis: card (union (rng Fy)) = Sum XFS
percases ( k = 0 or k > 0 ) ;
supposeA7: k = 0 ; ::_thesis: card (union (rng Fy)) = Sum XFS
then len XFS = 1 by A4, A5;
then A8: XFS = <%(XFS . 0)%> by AFINSQ_1:34;
XFS . 0 is Element of INT by INT_1:def_2;
then A9: addint "**" XFS = XFS . 0 by A8, AFINSQ_2:37;
0 in dom XFS by A4, A5, A7, CARD_1:49, TARSKI:def_1;
then A10: XFS . 0 = ((- 1) |^ 0) * (Card_Intersection (Fy,(0 + 1))) by A6;
consider x being set such that
A11: dom Fy = {x} by A3, A4, A7, CARD_2:42;
A12: rng Fy = {(Fy . x)} by A11, FUNCT_1:4;
then A13: union (rng Fy) = Fy . x by ZFMISC_1:25;
( (- 1) |^ 0 = 1 & Fy = x .--> (Fy . x) ) by A11, A12, FUNCOP_1:9, NEWTON:4;
then XFS . 0 = card (union (rng Fy)) by Th46, A13, A10;
hence card (union (rng Fy)) = Sum XFS by A9, AFINSQ_2:50; ::_thesis: verum
end;
supposeA14: k > 0 ; ::_thesis: card (union (rng Fy)) = Sum XFS
consider x being set such that
A15: x in dom Fy by A3, A4, CARD_1:27, XBOOLE_0:def_1;
set Xx = X \ {x};
A16: card (X \ {x}) = k by A3, A4, A15, STIRL2_1:55;
set FyX = Fy | (X \ {x});
reconsider urngFyX = union (rng (Fy | (X \ {x}))) as finite set ;
set Fyx = Fy . x;
set I = Intersect ((Fy | (X \ {x})),((dom (Fy | (X \ {x}))) --> (Fy . x)));
consider XFyX being XFinSequence of such that
A17: dom XFyX = card (X \ {x}) and
A18: for n being Nat st n in dom XFyX holds
XFyX . n = ((- 1) |^ n) * (Card_Intersection ((Fy | (X \ {x})),(n + 1))) by Th55;
urngFyX /\ (Fy . x) = union (rng (Intersect ((Fy | (X \ {x})),((dom (Fy | (X \ {x}))) --> (Fy . x))))) by Th50;
then reconsider urngI = union (rng (Intersect ((Fy | (X \ {x})),((dom (Fy | (X \ {x}))) --> (Fy . x))))) as finite set ;
consider XI being XFinSequence of such that
A19: dom XI = card (X \ {x}) and
A20: for n being Nat st n in dom XI holds
XI . n = ((- 1) |^ n) * (Card_Intersection ((Intersect ((Fy | (X \ {x})),((dom (Fy | (X \ {x}))) --> (Fy . x)))),(n + 1))) by Th55;
set XI1 = (- 1) (#) XI;
reconsider XI1 = (- 1) (#) XI as XFinSequence of ;
reconsider XcF = <%(card (Fy . x))%>, X0 = <%0%> as XFinSequence of ;
reconsider F1 = <%(card (Fy . x))%> ^ XI1, F2 = XFyX ^ <%0%> as XFinSequence of ;
A21: card (X \ {x}) = k by A3, A4, A15, STIRL2_1:55;
0 is Element of INT by INT_1:def_2;
then A22: addint "**" X0 = 0 by AFINSQ_2:37;
card (Fy . x) is Element of INT by INT_1:def_2;
then A23: addint "**" XcF = card (Fy . x) by AFINSQ_2:37;
A24: (- 1) * (Sum XI) = Sum XI1 by AFINSQ_2:64;
A25: addint "**" F1 = addint . ((card (Fy . x)),(addint "**" XI1)) by A23, AFINSQ_2:42
.= (card (Fy . x)) + (addint "**" XI1) by BINOP_2:def_20
.= (card (Fy . x)) + (Sum XI1) by AFINSQ_2:50 ;
A26: addint "**" F2 = addint . ((addint "**" XFyX),0) by A22, AFINSQ_2:42
.= (addint "**" XFyX) + 0 by BINOP_2:def_20
.= Sum XFyX by AFINSQ_2:50 ;
A27: Sum (F1 ^ F2) = (Sum F1) + (Sum F2) by AFINSQ_2:55
.= (addint "**" F1) + (Sum F2) by AFINSQ_2:50
.= ((card (Fy . x)) + ((- 1) * (Sum XI))) + (Sum XFyX) by A24, A25, A26, AFINSQ_2:50 ;
A28: urngFyX \/ (Fy . x) = urngFy by A3, A15, Th54;
A29: urngFyX /\ (Fy . x) = urngI by Th50;
A30: dom (Fy | (X \ {x})) = X \ {x} by A3, RELAT_1:62;
then dom (Intersect ((Fy | (X \ {x})),((dom (Fy | (X \ {x}))) --> (Fy . x)))) = X \ {x} by Th49;
then A31: card urngI = Sum XI by A2, A19, A20, A21;
( len <%(card (Fy . x))%> = 1 & len XI1 = card (X \ {x}) ) by A19, AFINSQ_1:33, VALUED_1:def_5;
then A32: len F1 = k + 1 by A16, AFINSQ_1:17;
A33: for n being Nat st n in dom XFS holds
XFS . n = addint . ((F1 . n),(F2 . n))
proof
let n be Nat; ::_thesis: ( n in dom XFS implies XFS . n = addint . ((F1 . n),(F2 . n)) )
assume A34: n in dom XFS ; ::_thesis: XFS . n = addint . ((F1 . n),(F2 . n))
reconsider N = n as Element of NAT by ORDINAL1:def_12;
percases ( n = 0 or n > 0 ) ;
supposeA35: n = 0 ; ::_thesis: XFS . n = addint . ((F1 . n),(F2 . n))
( 0 in k & k = dom XFyX ) by A3, A4, A14, A15, A17, NAT_1:44, STIRL2_1:55;
then A36: ( F2 . 0 = XFyX . 0 & XFyX . 0 = ((- 1) |^ 0) * (Card_Intersection ((Fy | (X \ {x})),(0 + 1))) ) by A18, AFINSQ_1:def_3;
( F1 . 0 = card (Fy . x) & (- 1) |^ 0 = 1 ) by AFINSQ_1:35, NEWTON:4;
then A37: addint . ((F1 . 0),(F2 . 0)) = (card (Fy . x)) + (Card_Intersection ((Fy | (X \ {x})),(0 + 1))) by A36, BINOP_2:def_20;
A38: (- 1) |^ 0 = 1 by NEWTON:4;
XFS . 0 = ((- 1) |^ 0) * (Card_Intersection (Fy,(0 + 1))) by A6, A34, A35;
hence XFS . n = addint . ((F1 . n),(F2 . n)) by A3, A15, A35, A37, A38, Th48; ::_thesis: verum
end;
supposeA39: n > 0 ; ::_thesis: XFS . n = addint . ((F1 . n),(F2 . n))
then reconsider n1 = n - 1 as Element of NAT by NAT_1:20;
A40: len <%(card (Fy . x))%> = 1 by AFINSQ_1:33;
A41: card (X \ {x}) = k by A3, A4, A15, STIRL2_1:55;
A42: n < k + 1 by A4, A5, A34, NAT_1:44;
then A43: n <= k by NAT_1:13;
A44: n1 < n1 + 1 by NAT_1:13;
then n1 < k by A43, XXREAL_0:2;
then n1 in dom XI by A19, A41, NAT_1:44;
then A45: ( XI1 . n1 = (- 1) * (XI . n1) & XI . n1 = ((- 1) |^ n1) * (Card_Intersection ((Intersect ((Fy | (X \ {x})),((dom (Fy | (X \ {x}))) --> (Fy . x)))),(n1 + 1))) ) by A20, VALUED_1:6;
0 + 1 <= n by A44, NAT_1:13;
then F1 . n = ((- 1) * ((- 1) |^ n1)) * (Card_Intersection ((Intersect ((Fy | (X \ {x})),((dom (Fy | (X \ {x}))) --> (Fy . x)))),(n1 + 1))) by A32, A42, A40, A45, AFINSQ_1:19;
then A46: F1 . n = ((- 1) |^ (n1 + 1)) * (Card_Intersection ((Intersect ((Fy | (X \ {x})),((dom (Fy | (X \ {x}))) --> (Fy . x)))),(n1 + 1))) by NEWTON:6;
A47: XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) by A6, A34;
Card_Intersection (Fy,(n + 1)) = (Card_Intersection ((Fy | (X \ {x})),(n + 1))) + (Card_Intersection ((Intersect ((Fy | (X \ {x})),((dom (Fy | (X \ {x}))) --> (Fy . x)))),N)) by A3, A15, A30, A39, Th53;
then A48: XFS . n = (((- 1) |^ n) * (Card_Intersection ((Fy | (X \ {x})),(n + 1)))) + (((- 1) |^ n) * (Card_Intersection ((Intersect ((Fy | (X \ {x})),((dom (Fy | (X \ {x}))) --> (Fy . x)))),N))) by A47;
percases ( n < k or n = k ) by A43, XXREAL_0:1;
suppose n < k ; ::_thesis: XFS . n = addint . ((F1 . n),(F2 . n))
then A49: n in k by NAT_1:44;
card (X \ {x}) = k by A3, A4, A15, STIRL2_1:55;
then ( XFyX . n = ((- 1) |^ n) * (Card_Intersection ((Fy | (X \ {x})),(n + 1))) & F2 . n = XFyX . n ) by A17, A18, A49, AFINSQ_1:def_3;
hence XFS . n = addint . ((F1 . n),(F2 . n)) by A48, A46, BINOP_2:def_20; ::_thesis: verum
end;
supposeA50: n = k ; ::_thesis: XFS . n = addint . ((F1 . n),(F2 . n))
then n = card (X \ {x}) by A3, A4, A15, STIRL2_1:55;
then n + 1 > card (X \ {x}) by NAT_1:13;
then A51: Card_Intersection ((Fy | (X \ {x})),(n + 1)) = 0 by A30, Th43;
n = len XFyX by A3, A4, A15, A17, A50, STIRL2_1:55;
then F2 . n = 0 by AFINSQ_1:36;
hence XFS . n = addint . ((F1 . n),(F2 . n)) by A48, A46, A51, BINOP_2:def_20; ::_thesis: verum
end;
end;
end;
end;
end;
card urngFyX = Sum XFyX by A2, A30, A17, A18, A21;
then A52: card urngFy = ((Sum XFyX) + (card (Fy . x))) - (Sum XI) by A31, A28, A29, CARD_2:45;
A53: len <%0%> = 1 by AFINSQ_1:33;
len XFyX = card (X \ {x}) by A17;
then A54: len F2 = k + 1 by A53, A16, AFINSQ_1:17;
A55: len XFS = k + 1 by A4, A5;
Sum XFS = addint "**" XFS by AFINSQ_2:50
.= addint "**" (F1 ^ F2) by A32, A54, A33, A55, AFINSQ_2:46
.= Sum (F1 ^ F2) by AFINSQ_2:50 ;
hence card (union (rng Fy)) = Sum XFS by A27, A52; ::_thesis: verum
end;
end;
end;
A56: S1[ 0 ]
proof
let Fy be finite-yielding Function; ::_thesis: for X being finite set st dom Fy = X & card X = 0 holds
for XFS being XFinSequence of st dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) holds
card (union (rng Fy)) = Sum XFS
let X be finite set ; ::_thesis: ( dom Fy = X & card X = 0 implies for XFS being XFinSequence of st dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) holds
card (union (rng Fy)) = Sum XFS )
assume that
A57: dom Fy = X and
A58: card X = 0 ; ::_thesis: for XFS being XFinSequence of st dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) holds
card (union (rng Fy)) = Sum XFS
dom Fy = {} by A57, A58;
then A59: rng Fy = {} by RELAT_1:42;
let XFS be XFinSequence of ; ::_thesis: ( dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) implies card (union (rng Fy)) = Sum XFS )
assume that
A60: dom XFS = card X and
for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ; ::_thesis: card (union (rng Fy)) = Sum XFS
len XFS = 0 by A58, A60;
then addint "**" XFS = the_unity_wrt addint by AFINSQ_2:def_8
.= 0 by BINOP_2:4 ;
hence card (union (rng Fy)) = Sum XFS by A59, A58, AFINSQ_2:50, ZFMISC_1:2; ::_thesis: verum
end;
for k being Nat holds S1[k] from NAT_1:sch_2(A56, A1);
hence for Fy being finite-yielding Function
for X being finite set st dom Fy = X holds
for XFS being XFinSequence of st dom XFS = card X & ( for n being Nat st n in dom XFS holds
XFS . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) ) holds
card (union (rng Fy)) = Sum XFS ; ::_thesis: verum
end;
theorem Th57: :: CARD_FIN:57
for Fy being finite-yielding Function
for X being finite set
for n, k being Nat st dom Fy = X & ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,k,x,y) holds
card (Intersection (Fy,f,x)) = n ) ) holds
Card_Intersection (Fy,k) = n * ((card X) choose k)
proof
let Fy be finite-yielding Function; ::_thesis: for X being finite set
for n, k being Nat st dom Fy = X & ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,k,x,y) holds
card (Intersection (Fy,f,x)) = n ) ) holds
Card_Intersection (Fy,k) = n * ((card X) choose k)
let X be finite set ; ::_thesis: for n, k being Nat st dom Fy = X & ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,k,x,y) holds
card (Intersection (Fy,f,x)) = n ) ) holds
Card_Intersection (Fy,k) = n * ((card X) choose k)
let n, k be Nat; ::_thesis: ( dom Fy = X & ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,k,x,y) holds
card (Intersection (Fy,f,x)) = n ) ) implies Card_Intersection (Fy,k) = n * ((card X) choose k) )
assume A1: X = dom Fy ; ::_thesis: ( for x, y being set holds
( not x <> y or ex f being Function st
( f in Choose (X,k,x,y) & not card (Intersection (Fy,f,x)) = n ) ) or Card_Intersection (Fy,k) = n * ((card X) choose k) )
assume ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,k,x,y) holds
card (Intersection (Fy,f,x)) = n ) ) ; ::_thesis: Card_Intersection (Fy,k) = n * ((card X) choose k)
then consider x, y being set such that
A2: x <> y and
A3: for f being Function st f in Choose (X,k,x,y) holds
card (Intersection (Fy,f,x)) = n ;
set Ch = Choose (X,k,x,y);
consider P being Function of (card (Choose (X,k,x,y))),(Choose (X,k,x,y)) such that
A4: P is one-to-one by Lm2;
consider XFS being XFinSequence of such that
A5: dom XFS = dom P and
A6: for z being set
for f being Function st z in dom XFS & f = P . z holds
XFS . z = card (Intersection (Fy,f,x)) and
A7: Card_Intersection (Fy,k) = Sum XFS by A1, A2, A4, Def4;
for z being set st z in dom XFS holds
XFS . z = n
proof
let z be set ; ::_thesis: ( z in dom XFS implies XFS . z = n )
assume A8: z in dom XFS ; ::_thesis: XFS . z = n
A9: P . z in rng P by A5, A8, FUNCT_1:def_3;
then consider f being Function of X,{x,y} such that
A10: f = P . z and
card (f " {x}) = k by Def1;
XFS . z = card (Intersection (Fy,f,x)) by A6, A8, A10;
hence XFS . z = n by A3, A9, A10; ::_thesis: verum
end;
then A11: XFS = (dom XFS) --> n by FUNCOP_1:11;
then A12: rng XFS c= {n} by FUNCOP_1:13;
( Choose (X,k,x,y) = {} implies card (Choose (X,k,x,y)) = {} ) ;
then A13: dom P = card (Choose (X,k,x,y)) by FUNCT_2:def_1;
n in {n} by TARSKI:def_1;
then ( {n} c= {0,n} & XFS " {n} = dom P ) by A5, A11, FUNCOP_1:14, ZFMISC_1:7;
then Sum XFS = n * (card (card (Choose (X,k,x,y)))) by A12, A13, AFINSQ_2:68, XBOOLE_1:1;
hence Card_Intersection (Fy,k) = n * ((card X) choose k) by A2, A7, Th16; ::_thesis: verum
end;
theorem Th58: :: CARD_FIN:58
for Fy being finite-yielding Function
for X being finite set st dom Fy = X holds
for XF being XFinSequence of st dom XF = card X & ( for n being Nat st n in dom XF holds
ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,(n + 1),x,y) holds
card (Intersection (Fy,f,x)) = XF . n ) ) ) holds
ex F being XFinSequence of st
( dom F = card X & card (union (rng Fy)) = Sum F & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * (XF . n)) * ((card X) choose (n + 1)) ) )
proof
let Fy be finite-yielding Function; ::_thesis: for X being finite set st dom Fy = X holds
for XF being XFinSequence of st dom XF = card X & ( for n being Nat st n in dom XF holds
ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,(n + 1),x,y) holds
card (Intersection (Fy,f,x)) = XF . n ) ) ) holds
ex F being XFinSequence of st
( dom F = card X & card (union (rng Fy)) = Sum F & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * (XF . n)) * ((card X) choose (n + 1)) ) )
let X be finite set ; ::_thesis: ( dom Fy = X implies for XF being XFinSequence of st dom XF = card X & ( for n being Nat st n in dom XF holds
ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,(n + 1),x,y) holds
card (Intersection (Fy,f,x)) = XF . n ) ) ) holds
ex F being XFinSequence of st
( dom F = card X & card (union (rng Fy)) = Sum F & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * (XF . n)) * ((card X) choose (n + 1)) ) ) )
assume A1: dom Fy = X ; ::_thesis: for XF being XFinSequence of st dom XF = card X & ( for n being Nat st n in dom XF holds
ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,(n + 1),x,y) holds
card (Intersection (Fy,f,x)) = XF . n ) ) ) holds
ex F being XFinSequence of st
( dom F = card X & card (union (rng Fy)) = Sum F & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * (XF . n)) * ((card X) choose (n + 1)) ) )
let XF be XFinSequence of ; ::_thesis: ( dom XF = card X & ( for n being Nat st n in dom XF holds
ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,(n + 1),x,y) holds
card (Intersection (Fy,f,x)) = XF . n ) ) ) implies ex F being XFinSequence of st
( dom F = card X & card (union (rng Fy)) = Sum F & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * (XF . n)) * ((card X) choose (n + 1)) ) ) )
assume A2: ( dom XF = card X & ( for n being Nat st n in dom XF holds
ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,(n + 1),x,y) holds
card (Intersection (Fy,f,x)) = XF . n ) ) ) ) ; ::_thesis: ex F being XFinSequence of st
( dom F = card X & card (union (rng Fy)) = Sum F & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * (XF . n)) * ((card X) choose (n + 1)) ) )
defpred S1[ set , set ] means for n being Nat st n = $1 holds
$2 = (((- 1) |^ n) * (XF . n)) * ((card X) choose (n + 1));
A3: for x being set st x in card X holds
ex y being set st
( y in INT & S1[x,y] )
proof
A4: card X is Subset of NAT by STIRL2_1:8;
let x be set ; ::_thesis: ( x in card X implies ex y being set st
( y in INT & S1[x,y] ) )
assume x in card X ; ::_thesis: ex y being set st
( y in INT & S1[x,y] )
then reconsider x9 = x as Element of NAT by A4;
reconsider xx = ((- 1) |^ x9) * (XF . x9) as Integer ;
reconsider ch = (card X) choose (x9 + 1) as Integer ;
take xx * ch ; ::_thesis: ( xx * ch in INT & S1[x,xx * ch] )
thus ( xx * ch in INT & S1[x,xx * ch] ) ; ::_thesis: verum
end;
consider F being Function of (card X),INT such that
A5: for x being set st x in card X holds
S1[x,F . x] from FUNCT_2:sch_1(A3);
A6: dom F = card X by FUNCT_2:def_1;
then reconsider F = F as XFinSequence by AFINSQ_1:5;
reconsider F = F as XFinSequence of ;
take F ; ::_thesis: ( dom F = card X & card (union (rng Fy)) = Sum F & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * (XF . n)) * ((card X) choose (n + 1)) ) )
for n being Nat st n in dom F holds
F . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1)))
proof
let n be Nat; ::_thesis: ( n in dom F implies F . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) )
assume A7: n in dom F ; ::_thesis: F . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1)))
ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,(n + 1),x,y) holds
card (Intersection (Fy,f,x)) = XF . n ) ) by A2, A6, A7;
then A8: Card_Intersection (Fy,(n + 1)) = (XF . n) * ((card X) choose (n + 1)) by A1, Th57;
F . n = (((- 1) |^ n) * (XF . n)) * ((card X) choose (n + 1)) by A5, A6, A7;
hence F . n = ((- 1) |^ n) * (Card_Intersection (Fy,(n + 1))) by A8; ::_thesis: verum
end;
hence ( dom F = card X & card (union (rng Fy)) = Sum F & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * (XF . n)) * ((card X) choose (n + 1)) ) ) by A1, A5, A6, Th56; ::_thesis: verum
end;
Lm3: for X, Y being finite set st not X is empty & not Y is empty holds
ex F being XFinSequence of st
( dom F = card Y & ((card Y) |^ (card X)) - (Sum F) = card { f where f is Function of X,Y : f is onto } & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * ((card Y) choose (n + 1))) * ((((card Y) - n) - 1) |^ (card X)) ) )
proof
let X, Y be finite set ; ::_thesis: ( not X is empty & not Y is empty implies ex F being XFinSequence of st
( dom F = card Y & ((card Y) |^ (card X)) - (Sum F) = card { f where f is Function of X,Y : f is onto } & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * ((card Y) choose (n + 1))) * ((((card Y) - n) - 1) |^ (card X)) ) ) )
assume that
A1: not X is empty and
A2: not Y is empty ; ::_thesis: ex F being XFinSequence of st
( dom F = card Y & ((card Y) |^ (card X)) - (Sum F) = card { f where f is Function of X,Y : f is onto } & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * ((card Y) choose (n + 1))) * ((((card Y) - n) - 1) |^ (card X)) ) )
defpred S1[ set , set ] means for n being Nat st n = $1 holds
$2 = (((card Y) - n) - 1) |^ (card X);
A3: for x being set st x in card Y holds
ex y being set st
( y in NAT & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in card Y implies ex y being set st
( y in NAT & S1[x,y] ) )
assume A4: x in card Y ; ::_thesis: ex y being set st
( y in NAT & S1[x,y] )
card Y is Subset of NAT by STIRL2_1:8;
then reconsider n = x as Element of NAT by A4;
n < card Y by A4, NAT_1:44;
then n + 1 <= card Y by NAT_1:13;
then reconsider k = (card Y) - (n + 1) as Element of NAT by NAT_1:21;
S1[n,k |^ (card X)] ;
hence ex y being set st
( y in NAT & S1[x,y] ) ; ::_thesis: verum
end;
consider XF being Function of (card Y),NAT such that
A5: for x being set st x in card Y holds
S1[x,XF . x] from FUNCT_2:sch_1(A3);
set Onto = { f where f is Function of X,Y : f is onto } ;
deffunc H1( set ) -> set = { f where f is Function of X,Y : not $1 in rng f } ;
A6: for x being set st x in Y holds
H1(x) in bool (Funcs (X,Y))
proof
let x be set ; ::_thesis: ( x in Y implies H1(x) in bool (Funcs (X,Y)) )
assume A7: x in Y ; ::_thesis: H1(x) in bool (Funcs (X,Y))
H1(x) c= Funcs (X,Y)
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in H1(x) or y in Funcs (X,Y) )
assume y in H1(x) ; ::_thesis: y in Funcs (X,Y)
then ex f being Function of X,Y st
( y = f & not x in rng f ) ;
hence y in Funcs (X,Y) by A7, FUNCT_2:8; ::_thesis: verum
end;
hence H1(x) in bool (Funcs (X,Y)) ; ::_thesis: verum
end;
consider Fy9 being Function of Y,(bool (Funcs (X,Y))) such that
A8: for x being set st x in Y holds
Fy9 . x = H1(x) from FUNCT_2:sch_2(A6);
for y being set st y in dom Fy9 holds
Fy9 . y is finite
proof
let y be set ; ::_thesis: ( y in dom Fy9 implies Fy9 . y is finite )
assume y in dom Fy9 ; ::_thesis: Fy9 . y is finite
then Fy9 . y in rng Fy9 by FUNCT_1:def_3;
hence Fy9 . y is finite ; ::_thesis: verum
end;
then reconsider Fy = Fy9 as finite-yielding Function by FINSET_1:def_4;
union (rng Fy9) c= union (bool (Funcs (X,Y))) by ZFMISC_1:77;
then A9: union (rng Fy) c= Funcs (X,Y) by ZFMISC_1:81;
reconsider u = union (rng Fy) as finite set ;
A10: dom XF = card Y by FUNCT_2:def_1;
then reconsider XF = XF as XFinSequence by AFINSQ_1:5;
reconsider XF = XF as XFinSequence of ;
A11: for n being Nat st n in dom XF holds
ex x, y being set st
( x <> y & ( for f being Function st f in Choose (Y,(n + 1),x,y) holds
card (Intersection (Fy,f,x)) = XF . n ) )
proof
let n be Nat; ::_thesis: ( n in dom XF implies ex x, y being set st
( x <> y & ( for f being Function st f in Choose (Y,(n + 1),x,y) holds
card (Intersection (Fy,f,x)) = XF . n ) ) )
assume A12: n in dom XF ; ::_thesis: ex x, y being set st
( x <> y & ( for f being Function st f in Choose (Y,(n + 1),x,y) holds
card (Intersection (Fy,f,x)) = XF . n ) )
take 0 ; ::_thesis: ex y being set st
( 0 <> y & ( for f being Function st f in Choose (Y,(n + 1),0,y) holds
card (Intersection (Fy,f,0)) = XF . n ) )
take 1 ; ::_thesis: ( 0 <> 1 & ( for f being Function st f in Choose (Y,(n + 1),0,1) holds
card (Intersection (Fy,f,0)) = XF . n ) )
thus 0 <> 1 ; ::_thesis: for f being Function st f in Choose (Y,(n + 1),0,1) holds
card (Intersection (Fy,f,0)) = XF . n
let f9 be Function; ::_thesis: ( f9 in Choose (Y,(n + 1),0,1) implies card (Intersection (Fy,f9,0)) = XF . n )
assume f9 in Choose (Y,(n + 1),0,1) ; ::_thesis: card (Intersection (Fy,f9,0)) = XF . n
then consider f being Function of Y,{0,1} such that
A13: f = f9 and
A14: card (f " {0}) = n + 1 by Def1;
A15: Intersection (Fy,f,0) c= Funcs (X,(Y \ (f " {0})))
proof
let z be set ; :: according to TARSKI:def_3 ::_thesis: ( not z in Intersection (Fy,f,0) or z in Funcs (X,(Y \ (f " {0}))) )
assume A16: z in Intersection (Fy,f,0) ; ::_thesis: z in Funcs (X,(Y \ (f " {0})))
0 in rng f by A14, CARD_1:27, FUNCT_1:72;
then consider x1 being set such that
A17: x1 in dom f and
f . x1 = 0 and
A18: z in Fy . x1 by A16, Th22;
z in H1(x1) by A8, A17, A18;
then consider g being Function of X,Y such that
A19: z = g and
not x1 in rng g ;
A20: rng g c= Y \ (f " {0})
proof
let gy be set ; :: according to TARSKI:def_3 ::_thesis: ( not gy in rng g or gy in Y \ (f " {0}) )
assume A21: gy in rng g ; ::_thesis: gy in Y \ (f " {0})
assume not gy in Y \ (f " {0}) ; ::_thesis: contradiction
then A22: gy in f " {0} by A21, XBOOLE_0:def_5;
then f . gy in {0} by FUNCT_1:def_7;
then A23: f . gy = 0 by TARSKI:def_1;
gy in dom f by A22, FUNCT_1:def_7;
then g in Fy . gy by A16, A19, A23, Def2;
then g in H1(gy) by A8, A21;
then ex h being Function of X,Y st
( g = h & not gy in rng h ) ;
hence contradiction by A21; ::_thesis: verum
end;
dom g = X by A17, FUNCT_2:def_1;
hence z in Funcs (X,(Y \ (f " {0}))) by A19, A20, FUNCT_2:def_2; ::_thesis: verum
end;
reconsider I = Intersection (Fy,f,0) as finite set ;
A24: card (Y \ (f " {0})) = (card Y) - (n + 1) by A14, CARD_2:44;
Funcs (X,(Y \ (f " {0}))) c= Intersection (Fy,f,0)
proof
let g9 be set ; :: according to TARSKI:def_3 ::_thesis: ( not g9 in Funcs (X,(Y \ (f " {0}))) or g9 in Intersection (Fy,f,0) )
assume g9 in Funcs (X,(Y \ (f " {0}))) ; ::_thesis: g9 in Intersection (Fy,f,0)
then consider g being Function such that
A25: g9 = g and
A26: dom g = X and
A27: rng g c= Y \ (f " {0}) by FUNCT_2:def_2;
reconsider gg = g as Function of X,Y by A26, A27, FUNCT_2:2, XBOOLE_1:1;
consider y being set such that
A28: y in f " {0} by A14, CARD_1:27, XBOOLE_0:def_1;
not y in rng g by A27, A28, XBOOLE_0:def_5;
then A29: gg in H1(y) ;
dom Fy = Y by FUNCT_2:def_1;
then A30: Fy9 . y in rng Fy9 by A28, FUNCT_1:def_3;
A31: for z being set st z in dom f & f . z = 0 holds
g in Fy . z
proof
let z be set ; ::_thesis: ( z in dom f & f . z = 0 implies g in Fy . z )
assume that
A32: z in dom f and
A33: f . z = 0 ; ::_thesis: g in Fy . z
f . z in {0} by A33, TARSKI:def_1;
then z in f " {0} by A32, FUNCT_1:def_7;
then A34: not z in rng gg by A27, XBOOLE_0:def_5;
Fy . z = H1(z) by A8, A32;
hence g in Fy . z by A34; ::_thesis: verum
end;
H1(y) = Fy9 . y by A8, A28;
then g in union (rng Fy) by A30, A29, TARSKI:def_4;
hence g9 in Intersection (Fy,f,0) by A25, A31, Def2; ::_thesis: verum
end;
then A35: Funcs (X,(Y \ (f " {0}))) = Intersection (Fy,f,0) by A15, XBOOLE_0:def_10;
now__::_thesis:_card_(Intersection_(Fy,f9,0))_=_XF_._n
percases ( Y \ (f " {0}) = {} or Y \ (f " {0}) <> {} ) ;
suppose Y \ (f " {0}) = {} ; ::_thesis: card (Intersection (Fy,f9,0)) = XF . n
then ( card I = 0 & (((card Y) - n) - 1) |^ (card X) = 0 ) by A1, A15, A24, CARD_1:27, NAT_1:14, NEWTON:11;
hence card (Intersection (Fy,f9,0)) = XF . n by A5, A10, A12, A13; ::_thesis: verum
end;
supposeA36: Y \ (f " {0}) <> {} ; ::_thesis: card (Intersection (Fy,f9,0)) = XF . n
XF . n = (((card Y) - n) - 1) |^ (card X) by A5, A10, A12;
hence card (Intersection (Fy,f9,0)) = XF . n by A13, A35, A24, A36, Th4; ::_thesis: verum
end;
end;
end;
hence card (Intersection (Fy,f9,0)) = XF . n ; ::_thesis: verum
end;
( dom XF = card Y & dom Fy = Y ) by FUNCT_2:def_1;
then consider F being XFinSequence of such that
A37: dom F = card Y and
A38: card (union (rng Fy)) = Sum F and
A39: for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * (XF . n)) * ((card Y) choose (n + 1)) by A11, Th58;
take F ; ::_thesis: ( dom F = card Y & ((card Y) |^ (card X)) - (Sum F) = card { f where f is Function of X,Y : f is onto } & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * ((card Y) choose (n + 1))) * ((((card Y) - n) - 1) |^ (card X)) ) )
thus dom F = card Y by A37; ::_thesis: ( ((card Y) |^ (card X)) - (Sum F) = card { f where f is Function of X,Y : f is onto } & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * ((card Y) choose (n + 1))) * ((((card Y) - n) - 1) |^ (card X)) ) )
A40: card ((Funcs (X,Y)) \ u) = (card (Funcs (X,Y))) - (card u) by A9, CARD_2:44;
A41: { f where f is Function of X,Y : f is onto } c= (Funcs (X,Y)) \ (union (rng Fy))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { f where f is Function of X,Y : f is onto } or x in (Funcs (X,Y)) \ (union (rng Fy)) )
assume x in { f where f is Function of X,Y : f is onto } ; ::_thesis: x in (Funcs (X,Y)) \ (union (rng Fy))
then consider f being Function of X,Y such that
A42: x = f and
A43: f is onto ;
assume A44: not x in (Funcs (X,Y)) \ (union (rng Fy)) ; ::_thesis: contradiction
f in Funcs (X,Y) by A2, FUNCT_2:8;
then f in union (rng Fy) by A42, A44, XBOOLE_0:def_5;
then consider Fyy being set such that
A45: f in Fyy and
A46: Fyy in rng Fy by TARSKI:def_4;
consider y being set such that
A47: y in dom Fy and
A48: Fy . y = Fyy by A46, FUNCT_1:def_3;
y in Y by A47, FUNCT_2:def_1;
then f in H1(y) by A8, A45, A48;
then A49: ex g being Function of X,Y st
( f = g & not y in rng g ) ;
y in Y by A47, FUNCT_2:def_1;
hence contradiction by A43, A49, FUNCT_2:def_3; ::_thesis: verum
end;
A50: (Funcs (X,Y)) \ (union (rng Fy)) c= { f where f is Function of X,Y : f is onto }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in (Funcs (X,Y)) \ (union (rng Fy)) or x in { f where f is Function of X,Y : f is onto } )
assume A51: x in (Funcs (X,Y)) \ (union (rng Fy)) ; ::_thesis: x in { f where f is Function of X,Y : f is onto }
consider f being Function such that
A52: x = f and
A53: ( dom f = X & rng f c= Y ) by A51, FUNCT_2:def_2;
reconsider f = f as Function of X,Y by A53, FUNCT_2:2;
assume not x in { f where f is Function of X,Y : f is onto } ; ::_thesis: contradiction
then not f is onto by A52;
then rng f <> Y by FUNCT_2:def_3;
then not Y c= rng f by XBOOLE_0:def_10;
then consider y being set such that
A54: y in Y and
A55: not y in rng f by TARSKI:def_3;
y in dom Fy9 by A54, FUNCT_2:def_1;
then Fy9 . y in rng Fy9 by FUNCT_1:def_3;
then A56: H1(y) in rng Fy9 by A8, A54;
f in H1(y) by A55;
then f in union (rng Fy) by A56, TARSKI:def_4;
hence contradiction by A51, A52, XBOOLE_0:def_5; ::_thesis: verum
end;
card (Funcs (X,Y)) = (card Y) |^ (card X) by A2, Th4;
hence card { f where f is Function of X,Y : f is onto } = ((card Y) |^ (card X)) - (Sum F) by A38, A50, A41, A40, XBOOLE_0:def_10; ::_thesis: for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * ((card Y) choose (n + 1))) * ((((card Y) - n) - 1) |^ (card X))
let n be Nat; ::_thesis: ( n in dom F implies F . n = (((- 1) |^ n) * ((card Y) choose (n + 1))) * ((((card Y) - n) - 1) |^ (card X)) )
assume A57: n in dom F ; ::_thesis: F . n = (((- 1) |^ n) * ((card Y) choose (n + 1))) * ((((card Y) - n) - 1) |^ (card X))
A58: F . n = (((- 1) |^ n) * (XF . n)) * ((card Y) choose (n + 1)) by A39, A57;
XF . n = (((card Y) - n) - 1) |^ (card X) by A5, A37, A57;
hence F . n = (((- 1) |^ n) * ((card Y) choose (n + 1))) * ((((card Y) - n) - 1) |^ (card X)) by A58; ::_thesis: verum
end;
theorem Th59: :: CARD_FIN:59
for X, Y being finite set st not X is empty & not Y is empty holds
ex F being XFinSequence of st
( dom F = (card Y) + 1 & Sum F = card { f where f is Function of X,Y : f is onto } & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * ((card Y) choose n)) * (((card Y) - n) |^ (card X)) ) )
proof
let X, Y be finite set ; ::_thesis: ( not X is empty & not Y is empty implies ex F being XFinSequence of st
( dom F = (card Y) + 1 & Sum F = card { f where f is Function of X,Y : f is onto } & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * ((card Y) choose n)) * (((card Y) - n) |^ (card X)) ) ) )
assume A1: ( not X is empty & not Y is empty ) ; ::_thesis: ex F being XFinSequence of st
( dom F = (card Y) + 1 & Sum F = card { f where f is Function of X,Y : f is onto } & ( for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * ((card Y) choose n)) * (((card Y) - n) |^ (card X)) ) )
reconsider c = (card Y) |^ (card X) as Element of INT by INT_1:def_2;
A2: len <%c%> = 1 by AFINSQ_1:33;
set Onto = { f where f is Function of X,Y : f is onto } ;
consider F being XFinSequence of such that
A3: dom F = card Y and
A4: ((card Y) |^ (card X)) - (Sum F) = card { f where f is Function of X,Y : f is onto } and
A5: for n being Nat st n in dom F holds
F . n = (((- 1) |^ n) * ((card Y) choose (n + 1))) * ((((card Y) - n) - 1) |^ (card X)) by A1, Lm3;
set F1 = (- 1) (#) F;
reconsider F1 = (- 1) (#) F as XFinSequence of ;
A6: ( dom F1 = dom F & dom F = card Y ) by A3, VALUED_1:def_5;
reconsider GF1 = <%c%> ^ F1 as XFinSequence of ;
take GF1 ; ::_thesis: ( dom GF1 = (card Y) + 1 & Sum GF1 = card { f where f is Function of X,Y : f is onto } & ( for n being Nat st n in dom GF1 holds
GF1 . n = (((- 1) |^ n) * ((card Y) choose n)) * (((card Y) - n) |^ (card X)) ) )
len F1 = card Y by A3, VALUED_1:def_5;
hence A7: dom GF1 = (card Y) + 1 by A2, AFINSQ_1:def_3; ::_thesis: ( Sum GF1 = card { f where f is Function of X,Y : f is onto } & ( for n being Nat st n in dom GF1 holds
GF1 . n = (((- 1) |^ n) * ((card Y) choose n)) * (((card Y) - n) |^ (card X)) ) )
(- 1) * (Sum F) = Sum F1 by AFINSQ_2:64;
then c - (Sum F) = c + (Sum F1)
.= addint . (c,(Sum F1)) by BINOP_2:def_20
.= addint . ((addint "**" <%c%>),(Sum F1)) by AFINSQ_2:37
.= addint . ((addint "**" <%c%>),(addint "**" F1)) by AFINSQ_2:50
.= addint "**" GF1 by AFINSQ_2:42
.= Sum GF1 by AFINSQ_2:50 ;
hence Sum GF1 = card { f where f is Function of X,Y : f is onto } by A4; ::_thesis: for n being Nat st n in dom GF1 holds
GF1 . n = (((- 1) |^ n) * ((card Y) choose n)) * (((card Y) - n) |^ (card X))
let n be Nat; ::_thesis: ( n in dom GF1 implies GF1 . n = (((- 1) |^ n) * ((card Y) choose n)) * (((card Y) - n) |^ (card X)) )
assume A8: n in dom GF1 ; ::_thesis: GF1 . n = (((- 1) |^ n) * ((card Y) choose n)) * (((card Y) - n) |^ (card X))
now__::_thesis:_GF1_._n_=_(((-_1)_|^_n)_*_((card_Y)_choose_n))_*_(((card_Y)_-_n)_|^_(card_X))
percases ( n = 0 or n > 0 ) ;
supposeA9: n = 0 ; ::_thesis: GF1 . n = (((- 1) |^ n) * ((card Y) choose n)) * (((card Y) - n) |^ (card X))
then ( (- 1) |^ n = 1 & (card Y) choose n = 1 ) by NEWTON:4, NEWTON:19;
hence GF1 . n = (((- 1) |^ n) * ((card Y) choose n)) * (((card Y) - n) |^ (card X)) by A9, AFINSQ_1:35; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: GF1 . n = (((- 1) |^ n) * ((card Y) choose n)) * (((card Y) - n) |^ (card X))
then reconsider n1 = n - 1 as Element of NAT by NAT_1:20;
n < (card Y) + 1 by A7, A8, NAT_1:44;
then n1 + 1 <= card Y by NAT_1:13;
then n1 < card Y by NAT_1:13;
then A10: n1 in dom F1 by A6, NAT_1:44;
then A11: F . n1 = (((- 1) |^ n1) * ((card Y) choose (n1 + 1))) * ((((card Y) - n1) - 1) |^ (card X)) by A5, A6;
len <%c%> = 1 by AFINSQ_1:33;
then A12: GF1 . (n1 + 1) = F1 . n1 by A10, AFINSQ_1:def_3;
then A13: (- 1) * ((- 1) |^ n1) = (- 1) |^ n by NEWTON:6;
GF1 . (n1 + 1) = (- 1) * (F . n1) by A12, VALUED_1:6;
hence GF1 . n = (((- 1) |^ n) * ((card Y) choose n)) * (((card Y) - n) |^ (card X)) by A11, A13; ::_thesis: verum
end;
end;
end;
hence GF1 . n = (((- 1) |^ n) * ((card Y) choose n)) * (((card Y) - n) |^ (card X)) ; ::_thesis: verum
end;
theorem :: CARD_FIN:60
for n, k being Nat st k <= n holds
ex F being XFinSequence of st
( n block k = (1 / (k !)) * (Sum F) & dom F = k + 1 & ( for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) )
proof
let n, k be Nat; ::_thesis: ( k <= n implies ex F being XFinSequence of st
( n block k = (1 / (k !)) * (Sum F) & dom F = k + 1 & ( for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) ) )
assume A1: k <= n ; ::_thesis: ex F being XFinSequence of st
( n block k = (1 / (k !)) * (Sum F) & dom F = k + 1 & ( for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) )
now__::_thesis:_ex_F_being_set_st_
(_n_block_k_=_(1_/_(k_!))_*_(Sum_F)_&_dom_F_=_k_+_1_&_(_for_m_being_Nat_st_m_in_dom_F_holds_
F_._m_=_(((-_1)_|^_m)_*_(k_choose_m))_*_((k_-_m)_|^_n)_)_)
percases ( ( n = 0 & k = 0 ) or ( n <> 0 & k = 0 ) or ( n <> 0 & k <> 0 ) or ( n = 0 & k <> 0 ) ) ;
supposeA2: ( n = 0 & k = 0 ) ; ::_thesis: ex F being set st
( n block k = (1 / (k !)) * (Sum F) & dom F = k + 1 & ( for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) )
reconsider I = 1 as Element of INT by INT_1:def_2;
set F = <%I%>;
take F = <%I%>; ::_thesis: ( n block k = (1 / (k !)) * (Sum F) & dom F = k + 1 & ( for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) )
addint "**" <%I%> = 1 by AFINSQ_2:37;
then Sum F = 1 by AFINSQ_2:50;
hence n block k = (1 / (k !)) * (Sum F) by A2, NEWTON:12, STIRL2_1:26; ::_thesis: ( dom F = k + 1 & ( for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) )
thus dom F = k + 1 by A2, AFINSQ_1:33; ::_thesis: for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n)
let m be Nat; ::_thesis: ( m in dom F implies F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) )
assume m in dom F ; ::_thesis: F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n)
then A3: m in {0} by AFINSQ_1:33, CARD_1:49;
then m = 0 by TARSKI:def_1;
then A4: (- 1) |^ m = 1 by NEWTON:4;
A5: (k - m) |^ n = 1 by A2, NEWTON:4;
A6: 0 choose 0 = 1 by NEWTON:19;
m = 0 by A3, TARSKI:def_1;
hence F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) by A2, A4, A5, A6, AFINSQ_1:34; ::_thesis: verum
end;
supposeA7: ( n <> 0 & k = 0 ) ; ::_thesis: ex Fi being XFinSequence of st
( n block k = (1 / (k !)) * (Sum Fi) & dom Fi = k + 1 & ( for m being Nat st m in dom Fi holds
Fi . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) )
set F = (k + 1) --> 0;
reconsider Fi = (k + 1) --> 0 as XFinSequence of ;
reconsider Fn = (k + 1) --> 0 as XFinSequence of ;
take Fi = Fi; ::_thesis: ( n block k = (1 / (k !)) * (Sum Fi) & dom Fi = k + 1 & ( for m being Nat st m in dom Fi holds
Fi . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) )
( rng ((k + 1) --> 0) c= {0} & {0} c= {0,0} ) by ENUMSET1:29, FUNCOP_1:13;
then Sum Fn = 0 * (card (Fn " {0})) by AFINSQ_2:68, XBOOLE_1:1;
hence ( n block k = (1 / (k !)) * (Sum Fi) & dom Fi = k + 1 ) by A7, FUNCOP_1:13, STIRL2_1:31; ::_thesis: for m being Nat st m in dom Fi holds
Fi . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n)
let m be Nat; ::_thesis: ( m in dom Fi implies Fi . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) )
assume A8: m in dom Fi ; ::_thesis: Fi . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n)
now__::_thesis:_(k_choose_m)_*_((k_-_m)_|^_n)_=_0
percases ( m = 0 or m > 0 ) ;
suppose m = 0 ; ::_thesis: (k choose m) * ((k - m) |^ n) = 0
then (k - m) |^ n = 0 by A7, NAT_1:14, NEWTON:11;
hence (k choose m) * ((k - m) |^ n) = 0 ; ::_thesis: verum
end;
suppose m > 0 ; ::_thesis: (k choose m) * ((k - m) |^ n) = 0
then k choose m = 0 by A7, NEWTON:def_3;
hence (k choose m) * ((k - m) |^ n) = 0 ; ::_thesis: verum
end;
end;
end;
then A9: ((- 1) |^ m) * ((k choose m) * ((k - m) |^ n)) = 0 ;
m in k + 1 by A8, FUNCOP_1:13;
hence Fi . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) by A9, FUNCOP_1:7; ::_thesis: verum
end;
supposeA10: ( n <> 0 & k <> 0 ) ; ::_thesis: ex F being XFinSequence of st
( n block k = (1 / (k !)) * (Sum F) & dom F = k + 1 & ( for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) )
set Perm = { p where p is Function of k,k : p is Permutation of k } ;
card { p where p is Function of k,k : p is Permutation of k } = (card k) ! by Th8;
then reconsider Perm = { p where p is Function of k,k : p is Permutation of k } as finite set ;
reconsider Bloc = { f where f is Function of n,k : ( f is onto & f is "increasing ) } as finite set by STIRL2_1:24;
set Onto = { f where f is Function of n,k : f is onto } ;
defpred S1[ set , set ] means for p being Function of k,k
for f being Function of n,k st $1 = [p,f] holds
$2 = p * f;
reconsider N = n, K = k as non empty Subset of NAT by A10, STIRL2_1:8;
A11: card [:Perm,Bloc:] = (card Perm) * (card Bloc) by CARD_2:46;
A12: for x being set st x in [:Perm,Bloc:] holds
ex y being set st
( y in { f where f is Function of n,k : f is onto } & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in [:Perm,Bloc:] implies ex y being set st
( y in { f where f is Function of n,k : f is onto } & S1[x,y] ) )
assume x in [:Perm,Bloc:] ; ::_thesis: ex y being set st
( y in { f where f is Function of n,k : f is onto } & S1[x,y] )
then consider p9, f9 being set such that
A13: p9 in Perm and
A14: f9 in Bloc and
A15: x = [p9,f9] by ZFMISC_1:def_2;
consider f being Function of n,k such that
A16: f = f9 and
A17: ( f is onto & f is "increasing ) by A14;
A18: rng f = k by A17, FUNCT_2:def_3;
consider p being Function of k,k such that
A19: p = p9 and
A20: p is Permutation of k by A13;
reconsider pf = p * f as Function of n,k ;
take pf ; ::_thesis: ( pf in { f where f is Function of n,k : f is onto } & S1[x,pf] )
A21: dom p = k by A10, FUNCT_2:def_1;
rng p = k by A20, FUNCT_2:def_3;
then rng (p * f) = k by A18, A21, RELAT_1:28;
then pf is onto by FUNCT_2:def_3;
hence pf in { f where f is Function of n,k : f is onto } ; ::_thesis: S1[x,pf]
let p1 be Function of k,k; ::_thesis: for f being Function of n,k st x = [p1,f] holds
pf = p1 * f
let f1 be Function of n,k; ::_thesis: ( x = [p1,f1] implies pf = p1 * f1 )
assume A22: x = [p1,f1] ; ::_thesis: pf = p1 * f1
p1 = p by A15, A19, A22, XTUPLE_0:1;
hence pf = p1 * f1 by A15, A16, A22, XTUPLE_0:1; ::_thesis: verum
end;
consider FP being Function of [:Perm,Bloc:], { f where f is Function of n,k : f is onto } such that
A23: for x being set st x in [:Perm,Bloc:] holds
S1[x,FP . x] from FUNCT_2:sch_1(A12);
A24: FP is one-to-one
proof
let x1 be set ; :: according to FUNCT_1:def_4 ::_thesis: for b1 being set holds
( not x1 in dom FP or not b1 in dom FP or not FP . x1 = FP . b1 or x1 = b1 )
let x2 be set ; ::_thesis: ( not x1 in dom FP or not x2 in dom FP or not FP . x1 = FP . x2 or x1 = x2 )
assume that
A25: x1 in dom FP and
A26: x2 in dom FP and
A27: FP . x1 = FP . x2 ; ::_thesis: x1 = x2
consider p19, f19 being set such that
A28: p19 in Perm and
A29: f19 in Bloc and
A30: x1 = [p19,f19] by A25, ZFMISC_1:def_2;
consider p1 being Function of k,k such that
A31: p19 = p1 and
A32: p1 is Permutation of k by A28;
consider p29, f29 being set such that
A33: p29 in Perm and
A34: f29 in Bloc and
A35: x2 = [p29,f29] by A26, ZFMISC_1:def_2;
FP . x1 in rng FP by A25, FUNCT_1:def_3;
then FP . x1 in { f where f is Function of n,k : f is onto } ;
then consider fp being Function of N,K such that
A36: FP . x1 = fp and
A37: fp is onto ;
A38: rng fp = K by A37, FUNCT_2:def_3;
consider p2 being Function of k,k such that
A39: p29 = p2 and
A40: p2 is Permutation of k by A33;
consider f2 being Function of n,k such that
A41: f29 = f2 and
A42: ( f2 is onto & f2 is "increasing ) by A34;
rng fp = K by A37, FUNCT_2:def_3;
then reconsider p199 = p1, p299 = p2 as Permutation of (rng fp) by A32, A40;
consider f1 being Function of n,k such that
A43: f19 = f1 and
A44: ( f1 is onto & f1 is "increasing ) by A29;
reconsider f199 = f1, f299 = f2 as Function of N,K ;
A45: rng f2 = K by A42, FUNCT_2:def_3;
for l, m being Nat st l in rng f1 & m in rng f1 & l < m holds
min* (f1 " {l}) < min* (f1 " {m}) by A44, STIRL2_1:def_1;
then A46: f199 is "increasing by STIRL2_1:def_3;
for l, m being Nat st l in rng f2 & m in rng f2 & l < m holds
min* (f2 " {l}) < min* (f2 " {m}) by A42, STIRL2_1:def_1;
then A47: f299 is "increasing by STIRL2_1:def_3;
A48: fp = p199 * f199 by A23, A25, A30, A31, A43, A36;
A49: rng f1 = K by A44, FUNCT_2:def_3;
A50: fp = p299 * f299 by A23, A26, A27, A35, A39, A41, A36;
then p199 = p299 by A46, A47, A49, A45, A48, A38, STIRL2_1:65;
hence x1 = x2 by A30, A31, A43, A35, A39, A41, A46, A47, A49, A45, A48, A50, A38, STIRL2_1:65; ::_thesis: verum
end;
consider h being Function of n,k such that
A51: ( h is onto & h is "increasing ) by A1, A10, STIRL2_1:23;
{ f where f is Function of n,k : f is onto } c= rng FP
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { f where f is Function of n,k : f is onto } or x in rng FP )
assume x in { f where f is Function of n,k : f is onto } ; ::_thesis: x in rng FP
then consider f being Function of n,k such that
A52: f = x and
A53: f is onto ;
rng f = K by A53, FUNCT_2:def_3;
then consider I being Function of N,K, P being Permutation of K such that
A54: f = P * I and
A55: K = rng I and
A56: I is "increasing by STIRL2_1:63;
set p = P;
reconsider i = I as Function of n,k ;
for l, m being Nat st l in rng I & m in rng I & l < m holds
min* (I " {l}) < min* (I " {m}) by A56, STIRL2_1:def_3;
then A57: i is "increasing by STIRL2_1:def_1;
i is onto by A55, FUNCT_2:def_3;
then ( P in Perm & i in Bloc ) by A57;
then A58: [P,i] in [:Perm,Bloc:] by ZFMISC_1:def_2;
h in { f where f is Function of n,k : f is onto } by A51;
then A59: [P,i] in dom FP by A58, FUNCT_2:def_1;
FP . [P,i] = f by A23, A54, A58;
hence x in rng FP by A52, A59, FUNCT_1:def_3; ::_thesis: verum
end;
then A60: rng FP = { f where f is Function of n,k : f is onto } by XBOOLE_0:def_10;
h in { f where f is Function of n,k : f is onto } by A51;
then dom FP = [:Perm,Bloc:] by FUNCT_2:def_1;
then { f where f is Function of n,k : f is onto } ,[:Perm,Bloc:] are_equipotent by A24, A60, WELLORD2:def_4;
then A61: card { f where f is Function of n,k : f is onto } = (card Perm) * (card Bloc) by A11, CARD_1:5;
k ! > 0 by NEWTON:17;
then A62: ( ((k !) * (card Bloc)) / (k !) = (card Bloc) * ((k !) / (k !)) & (k !) / (k !) = 1 ) by XCMPLX_1:60, XCMPLX_1:74;
consider F being XFinSequence of such that
A63: dom F = (card k) + 1 and
A64: Sum F = card { f where f is Function of n,k : f is onto } and
A65: for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * ((card k) choose m)) * (((card k) - m) |^ (card n)) by A10, Th59;
take F = F; ::_thesis: ( n block k = (1 / (k !)) * (Sum F) & dom F = k + 1 & ( for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) )
card Perm = (card k) ! by Th8;
then Sum F = (k !) * (card Bloc) by A64, A61, CARD_1:def_2;
then n block k = ((Sum F) * 1) / (k !) by A62, STIRL2_1:def_2;
hence n block k = (1 / (k !)) * (Sum F) by XCMPLX_1:74; ::_thesis: ( dom F = k + 1 & ( for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) )
thus dom F = k + 1 by A63, CARD_1:def_2; ::_thesis: for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n)
A66: ( card k = k & card n = n ) by CARD_1:def_2;
let m be Nat; ::_thesis: ( m in dom F implies F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) )
assume m in dom F ; ::_thesis: F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n)
hence F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) by A65, A66; ::_thesis: verum
end;
suppose ( n = 0 & k <> 0 ) ; ::_thesis: ex F being XFinSequence of st
( n block k = (1 / (k !)) * (Sum F) & dom F = k + 1 & ( for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) )
hence ex F being XFinSequence of st
( n block k = (1 / (k !)) * (Sum F) & dom F = k + 1 & ( for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) ) by A1; ::_thesis: verum
end;
end;
end;
hence ex F being XFinSequence of st
( n block k = (1 / (k !)) * (Sum F) & dom F = k + 1 & ( for m being Nat st m in dom F holds
F . m = (((- 1) |^ m) * (k choose m)) * ((k - m) |^ n) ) ) ; ::_thesis: verum
end;
theorem Th61: :: CARD_FIN:61
for X1, Y1, X being finite set st ( Y1 is empty implies X1 is empty ) & X c= X1 holds
for F being Function of X1,Y1 st F is one-to-one & card X1 = card Y1 holds
((card X1) -' (card X)) ! = card { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) }
proof
let X1, Y1, X be finite set ; ::_thesis: ( ( Y1 is empty implies X1 is empty ) & X c= X1 implies for F being Function of X1,Y1 st F is one-to-one & card X1 = card Y1 holds
((card X1) -' (card X)) ! = card { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) } )
assume that
A1: ( Y1 is empty implies X1 is empty ) and
A2: X c= X1 ; ::_thesis: for F being Function of X1,Y1 st F is one-to-one & card X1 = card Y1 holds
((card X1) -' (card X)) ! = card { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) }
set XX = X1 \ X;
let F be Function of X1,Y1; ::_thesis: ( F is one-to-one & card X1 = card Y1 implies ((card X1) -' (card X)) ! = card { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) } )
assume that
A3: F is one-to-one and
A4: card X1 = card Y1 ; ::_thesis: ((card X1) -' (card X)) ! = card { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) }
deffunc H1( set ) -> set = F . $1;
defpred S1[ Function, set , set ] means ( $1 is one-to-one & rng ($1 | (X1 \ X)) = F .: (X1 \ X) );
reconsider FX = F .: (X1 \ X) as finite set ;
set F1 = { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : f is one-to-one } ;
A5: card (X1 \ X) = (card X1) - (card X) by A2, CARD_2:44;
A6: for f being Function of X1,Y1 st ( for x being set st x in X1 \ (X1 \ X) holds
H1(x) = f . x ) holds
( S1[f,X1,Y1] iff S1[f | (X1 \ X),X1 \ X,F .: (X1 \ X)] )
proof
let f be Function of X1,Y1; ::_thesis: ( ( for x being set st x in X1 \ (X1 \ X) holds
H1(x) = f . x ) implies ( S1[f,X1,Y1] iff S1[f | (X1 \ X),X1 \ X,F .: (X1 \ X)] ) )
assume A7: for x being set st x in X1 \ (X1 \ X) holds
H1(x) = f . x ; ::_thesis: ( S1[f,X1,Y1] iff S1[f | (X1 \ X),X1 \ X,F .: (X1 \ X)] )
thus ( S1[f,X1,Y1] implies S1[f | (X1 \ X),X1 \ X,F .: (X1 \ X)] ) by FUNCT_1:52; ::_thesis: ( S1[f | (X1 \ X),X1 \ X,F .: (X1 \ X)] implies S1[f,X1,Y1] )
thus ( S1[f | (X1 \ X),X1 \ X,F .: (X1 \ X)] implies S1[f,X1,Y1] ) ::_thesis: verum
proof
F is onto by A3, A4, STIRL2_1:60;
then A8: rng F = Y1 by FUNCT_2:def_3;
A9: ( rng (f | (X1 \ X)) = f .: (X1 \ X) & F .: ((X1 \ (X1 \ X)) \/ (X1 \ X)) = (F .: (X1 \ (X1 \ X))) \/ (F .: (X1 \ X)) ) by RELAT_1:115, RELAT_1:120;
A10: ( dom (F | (X1 \ (X1 \ X))) = (dom F) /\ (X1 \ (X1 \ X)) & dom F = X1 ) by A1, FUNCT_2:def_1, RELAT_1:61;
A11: ( dom (f | (X1 \ (X1 \ X))) = (dom f) /\ (X1 \ (X1 \ X)) & dom f = X1 ) by A1, FUNCT_2:def_1, RELAT_1:61;
now__::_thesis:_for_x_being_set_st_x_in_dom_(F_|_(X1_\_(X1_\_X)))_holds_
F_._x_=_(f_|_(X1_\_(X1_\_X)))_._x
A12: ( X1 \ (X1 \ X) = X /\ X1 & X /\ X1 = X ) by A2, XBOOLE_1:28, XBOOLE_1:48;
let x be set ; ::_thesis: ( x in dom (F | (X1 \ (X1 \ X))) implies F . x = (f | (X1 \ (X1 \ X))) . x )
assume A13: x in dom (F | (X1 \ (X1 \ X))) ; ::_thesis: F . x = (f | (X1 \ (X1 \ X))) . x
f . x = (f | (X1 \ (X1 \ X))) . x by A11, A10, A13, FUNCT_1:47;
hence F . x = (f | (X1 \ (X1 \ X))) . x by A7, A10, A13, A12; ::_thesis: verum
end;
then f | (X1 \ (X1 \ X)) = F | (X1 \ (X1 \ X)) by A11, A10, FUNCT_1:46;
then A14: rng (f | (X1 \ (X1 \ X))) = F .: (X1 \ (X1 \ X)) by RELAT_1:115;
A15: ( (X1 \ (X1 \ X)) \/ (X1 \ X) = X1 & rng (f | (X1 \ (X1 \ X))) = f .: (X1 \ (X1 \ X)) ) by RELAT_1:115, XBOOLE_1:45;
A16: ( X1 = dom F & X1 = dom f ) by A1, FUNCT_2:def_1;
A17: F .: (dom F) = rng F by RELAT_1:113;
assume A18: S1[f | (X1 \ X),X1 \ X,F .: (X1 \ X)] ; ::_thesis: S1[f,X1,Y1]
then rng (f | (X1 \ X)) = F .: (X1 \ X) ;
then F .: X1 = f .: X1 by A14, A15, A9, RELAT_1:120;
then rng F = rng f by A16, A17, RELAT_1:113;
then f is onto by A8, FUNCT_2:def_3;
hence S1[f,X1,Y1] by A4, A18, STIRL2_1:60; ::_thesis: verum
end;
end;
set F2 = { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) } ;
set S2 = { f where f is Function of X1,Y1 : ( S1[f,X1,Y1] & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X1 \ (X1 \ X) holds
f . x = H1(x) ) ) } ;
A19: ( X1 \ (X1 \ X) = X /\ X1 & X /\ X1 = X ) by A2, XBOOLE_1:28, XBOOLE_1:48;
A20: { f where f is Function of X1,Y1 : ( S1[f,X1,Y1] & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X1 \ (X1 \ X) holds
f . x = H1(x) ) ) } c= { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { f where f is Function of X1,Y1 : ( S1[f,X1,Y1] & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X1 \ (X1 \ X) holds
f . x = H1(x) ) ) } or x in { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) } )
assume x in { f where f is Function of X1,Y1 : ( S1[f,X1,Y1] & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X1 \ (X1 \ X) holds
f . x = H1(x) ) ) } ; ::_thesis: x in { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) }
then ex f being Function of X1,Y1 st
( x = f & S1[f,X1,Y1] & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X1 \ (X1 \ X) holds
f . x = H1(x) ) ) ;
hence x in { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) } by A19; ::_thesis: verum
end;
dom F = X1 by A1, FUNCT_2:def_1;
then X1 \ X,F .: (X1 \ X) are_equipotent by A3, CARD_1:33;
then A21: card (X1 \ X) = card (F .: (X1 \ X)) by CARD_1:5;
then ( card { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : f is one-to-one } = ((card (X1 \ X)) !) / (((card FX) -' (card (X1 \ X))) !) & (card FX) -' (card (X1 \ X)) = 0 ) by Th7, XREAL_1:232;
then A22: card { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : f is one-to-one } = ((card X1) -' (card X)) ! by A5, NEWTON:12, XREAL_0:def_2;
set S1 = { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : S1[f,X1 \ X,F .: (X1 \ X)] } ;
A23: for x being set st x in X1 \ (X1 \ X) holds
H1(x) in Y1
proof
A24: X1 = dom F by A1, FUNCT_2:def_1;
let x be set ; ::_thesis: ( x in X1 \ (X1 \ X) implies H1(x) in Y1 )
assume x in X1 \ (X1 \ X) ; ::_thesis: H1(x) in Y1
then F . x in rng F by A24, FUNCT_1:def_3;
hence H1(x) in Y1 ; ::_thesis: verum
end;
A25: { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : f is one-to-one } c= { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : S1[f,X1 \ X,F .: (X1 \ X)] }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : f is one-to-one } or x in { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : S1[f,X1 \ X,F .: (X1 \ X)] } )
assume x in { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : f is one-to-one } ; ::_thesis: x in { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : S1[f,X1 \ X,F .: (X1 \ X)] }
then consider f being Function of (X1 \ X),FX such that
A26: x = f and
A27: f is one-to-one ;
A28: f | (X1 \ X) = f ;
f is onto by A21, A27, STIRL2_1:60;
then rng f = FX by FUNCT_2:def_3;
hence x in { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : S1[f,X1 \ X,F .: (X1 \ X)] } by A26, A27, A28; ::_thesis: verum
end;
{ f where f is Function of (X1 \ X),(F .: (X1 \ X)) : S1[f,X1 \ X,F .: (X1 \ X)] } c= { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : f is one-to-one }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : S1[f,X1 \ X,F .: (X1 \ X)] } or x in { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : f is one-to-one } )
assume x in { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : S1[f,X1 \ X,F .: (X1 \ X)] } ; ::_thesis: x in { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : f is one-to-one }
then ex f being Function of (X1 \ X),FX st
( f = x & S1[f,X1 \ X,F .: (X1 \ X)] ) ;
hence x in { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : f is one-to-one } ; ::_thesis: verum
end;
then A29: { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : f is one-to-one } = { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : S1[f,X1 \ X,F .: (X1 \ X)] } by A25, XBOOLE_0:def_10;
A30: { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) } c= { f where f is Function of X1,Y1 : ( S1[f,X1,Y1] & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X1 \ (X1 \ X) holds
f . x = H1(x) ) ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) } or x in { f where f is Function of X1,Y1 : ( S1[f,X1,Y1] & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X1 \ (X1 \ X) holds
f . x = H1(x) ) ) } )
assume x in { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) } ; ::_thesis: x in { f where f is Function of X1,Y1 : ( S1[f,X1,Y1] & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X1 \ (X1 \ X) holds
f . x = H1(x) ) ) }
then consider f being Function of X1,Y1 such that
A31: x = f and
A32: f is one-to-one and
A33: rng (f | (X1 \ X)) c= F .: (X1 \ X) and
A34: for x being set st x in X holds
f . x = F . x ;
dom f = X1 by A1, FUNCT_2:def_1;
then X1 \ X,f .: (X1 \ X) are_equipotent by A32, CARD_1:33;
then card (X1 \ X) = card (f .: (X1 \ X)) by CARD_1:5;
then card FX = card (rng (f | (X1 \ X))) by A21, RELAT_1:115;
then rng (f | (X1 \ X)) = FX by A33, PRE_POLY:8;
hence x in { f where f is Function of X1,Y1 : ( S1[f,X1,Y1] & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X1 \ (X1 \ X) holds
f . x = H1(x) ) ) } by A19, A31, A32, A34; ::_thesis: verum
end;
A35: ( X1 \ X c= X1 & F .: (X1 \ X) c= Y1 ) ;
then X1 \ X c= dom F by A1, FUNCT_2:def_1;
then A36: ( F .: (X1 \ X) is empty implies X1 \ X is empty ) by RELAT_1:119;
card { f where f is Function of (X1 \ X),(F .: (X1 \ X)) : S1[f,X1 \ X,F .: (X1 \ X)] } = card { f where f is Function of X1,Y1 : ( S1[f,X1,Y1] & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X1 \ (X1 \ X) holds
f . x = H1(x) ) ) } from STIRL2_1:sch_3(A23, A35, A36, A6);
hence ((card X1) -' (card X)) ! = card { f where f is Function of X1,Y1 : ( f is one-to-one & rng (f | (X1 \ X)) c= F .: (X1 \ X) & ( for x being set st x in X holds
f . x = F . x ) ) } by A20, A30, A22, A29, XBOOLE_0:def_10; ::_thesis: verum
end;
Lm4: for X, Y being finite set
for F being Function of X,Y st dom F = X & F is one-to-one holds
ex XF being XFinSequence of st
( dom XF = card X & ((card X) !) - (Sum XF) = card { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } & ( for n being Nat st n in dom XF holds
XF . n = (((- 1) |^ n) * ((card X) !)) / ((n + 1) !) ) )
proof
let X, Y be finite set ; ::_thesis: for F being Function of X,Y st dom F = X & F is one-to-one holds
ex XF being XFinSequence of st
( dom XF = card X & ((card X) !) - (Sum XF) = card { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } & ( for n being Nat st n in dom XF holds
XF . n = (((- 1) |^ n) * ((card X) !)) / ((n + 1) !) ) )
let F be Function of X,Y; ::_thesis: ( dom F = X & F is one-to-one implies ex XF being XFinSequence of st
( dom XF = card X & ((card X) !) - (Sum XF) = card { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } & ( for n being Nat st n in dom XF holds
XF . n = (((- 1) |^ n) * ((card X) !)) / ((n + 1) !) ) ) )
assume that
A1: dom F = X and
A2: F is one-to-one ; ::_thesis: ex XF being XFinSequence of st
( dom XF = card X & ((card X) !) - (Sum XF) = card { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } & ( for n being Nat st n in dom XF holds
XF . n = (((- 1) |^ n) * ((card X) !)) / ((n + 1) !) ) )
deffunc H1( set ) -> set = { h where h is Function of X,(rng F) : ( h is one-to-one & h . $1 = F . $1 ) } ;
A3: for x being set st x in X holds
H1(x) in bool (Funcs (X,(rng F)))
proof
let x be set ; ::_thesis: ( x in X implies H1(x) in bool (Funcs (X,(rng F))) )
assume A4: x in X ; ::_thesis: H1(x) in bool (Funcs (X,(rng F)))
H1(x) c= Funcs (X,(rng F))
proof
let y be set ; :: according to TARSKI:def_3 ::_thesis: ( not y in H1(x) or y in Funcs (X,(rng F)) )
assume y in H1(x) ; ::_thesis: y in Funcs (X,(rng F))
then A5: ex h being Function of X,(rng F) st
( y = h & h is one-to-one & h . x = F . x ) ;
rng F <> {} by A1, A4, RELAT_1:42;
hence y in Funcs (X,(rng F)) by A5, FUNCT_2:8; ::_thesis: verum
end;
hence H1(x) in bool (Funcs (X,(rng F))) ; ::_thesis: verum
end;
consider Fy9 being Function of X,(bool (Funcs (X,(rng F)))) such that
A6: for x being set st x in X holds
Fy9 . x = H1(x) from FUNCT_2:sch_2(A3);
defpred S1[ set , set ] means for n, k being Nat st n = $1 & k = (card X) - (n + 1) holds
$2 = k ! ;
A7: for x being set st x in card X holds
ex y being set st
( y in NAT & S1[x,y] )
proof
let x be set ; ::_thesis: ( x in card X implies ex y being set st
( y in NAT & S1[x,y] ) )
assume A8: x in card X ; ::_thesis: ex y being set st
( y in NAT & S1[x,y] )
card X is Subset of NAT by STIRL2_1:8;
then reconsider n = x as Element of NAT by A8;
n < card X by A8, NAT_1:44;
then n + 1 <= card X by NAT_1:13;
then reconsider k = (card X) - (n + 1) as Element of NAT by NAT_1:21;
S1[n,k ! ] ;
hence ex y being set st
( y in NAT & S1[x,y] ) ; ::_thesis: verum
end;
consider XF being Function of (card X),NAT such that
A9: for x being set st x in card X holds
S1[x,XF . x] from FUNCT_2:sch_1(A7);
for y being set st y in dom Fy9 holds
Fy9 . y is finite
proof
let y be set ; ::_thesis: ( y in dom Fy9 implies Fy9 . y is finite )
assume y in dom Fy9 ; ::_thesis: Fy9 . y is finite
then Fy9 . y in rng Fy9 by FUNCT_1:def_3;
hence Fy9 . y is finite ; ::_thesis: verum
end;
then reconsider Fy = Fy9 as finite-yielding Function by FINSET_1:def_4;
reconsider rngF = rng F as finite set ;
A10: dom XF = card X by FUNCT_2:def_1;
then reconsider XF = XF as XFinSequence by AFINSQ_1:5;
reconsider XF = XF as XFinSequence of ;
A11: for n being Nat st n in dom XF holds
ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,(n + 1),x,y) holds
card (Intersection (Fy,f,x)) = XF . n ) )
proof
let n be Nat; ::_thesis: ( n in dom XF implies ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,(n + 1),x,y) holds
card (Intersection (Fy,f,x)) = XF . n ) ) )
assume A12: n in dom XF ; ::_thesis: ex x, y being set st
( x <> y & ( for f being Function st f in Choose (X,(n + 1),x,y) holds
card (Intersection (Fy,f,x)) = XF . n ) )
n < card X by A10, A12, NAT_1:44;
then A13: n + 1 <= card X by NAT_1:13;
then reconsider c = (card X) - (n + 1) as Element of NAT by NAT_1:21;
A14: (card X) -' (n + 1) = c by A13, XREAL_1:233;
take 0 ; ::_thesis: ex y being set st
( 0 <> y & ( for f being Function st f in Choose (X,(n + 1),0,y) holds
card (Intersection (Fy,f,0)) = XF . n ) )
take 1 ; ::_thesis: ( 0 <> 1 & ( for f being Function st f in Choose (X,(n + 1),0,1) holds
card (Intersection (Fy,f,0)) = XF . n ) )
thus 0 <> 1 ; ::_thesis: for f being Function st f in Choose (X,(n + 1),0,1) holds
card (Intersection (Fy,f,0)) = XF . n
let f9 be Function; ::_thesis: ( f9 in Choose (X,(n + 1),0,1) implies card (Intersection (Fy,f9,0)) = XF . n )
assume f9 in Choose (X,(n + 1),0,1) ; ::_thesis: card (Intersection (Fy,f9,0)) = XF . n
then consider f being Function of X,{0,1} such that
A15: f = f9 and
A16: card (f " {0}) = n + 1 by Def1;
reconsider f0 = f " {0} as finite set ;
set Xf0 = X \ f0;
set S = { h where h is Function of X,rngF : ( h is one-to-one & rng (h | (X \ f0)) c= F .: (X \ f0) & ( for x being set st x in f0 holds
h . x = F . x ) ) } ;
A17: Intersection (Fy,f,0) c= { h where h is Function of X,rngF : ( h is one-to-one & rng (h | (X \ f0)) c= F .: (X \ f0) & ( for x being set st x in f0 holds
h . x = F . x ) ) }
proof
assume not Intersection (Fy,f,0) c= { h where h is Function of X,rngF : ( h is one-to-one & rng (h | (X \ f0)) c= F .: (X \ f0) & ( for x being set st x in f0 holds
h . x = F . x ) ) } ; ::_thesis: contradiction
then consider z being set such that
A18: z in Intersection (Fy,f,0) and
A19: not z in { h where h is Function of X,rngF : ( h is one-to-one & rng (h | (X \ f0)) c= F .: (X \ f0) & ( for x being set st x in f0 holds
h . x = F . x ) ) } by TARSKI:def_3;
consider x9 being set such that
A20: x9 in f " {0} by A16, CARD_1:27, XBOOLE_0:def_1;
f . x9 in {0} by A20, FUNCT_1:def_7;
then A21: f . x9 = 0 by TARSKI:def_1;
x9 in dom f by A20, FUNCT_1:def_7;
then 0 in rng f by A21, FUNCT_1:def_3;
then consider x being set such that
A22: x in dom f and
f . x = 0 and
A23: z in Fy . x by A18, Th22;
z in H1(x) by A6, A22, A23;
then consider h being Function of X,(rng F) such that
A24: z = h and
A25: h is one-to-one and
h . x = F . x ;
A26: for x1 being set st x1 in f0 holds
h . x1 = F . x1
proof
let x1 be set ; ::_thesis: ( x1 in f0 implies h . x1 = F . x1 )
assume A27: x1 in f0 ; ::_thesis: h . x1 = F . x1
f . x1 in {0} by A27, FUNCT_1:def_7;
then A28: f . x1 = 0 by TARSKI:def_1;
( Fy9 . x1 = H1(x1) & x1 in dom f ) by A6, A27, FUNCT_1:def_7;
then h in H1(x1) by A18, A24, A28, Def2;
then ex h9 being Function of X,(rng F) st
( h = h9 & h9 is one-to-one & h9 . x1 = F . x1 ) ;
hence h . x1 = F . x1 ; ::_thesis: verum
end;
rng (h | (X \ f0)) c= F .: (X \ f0)
proof
assume not rng (h | (X \ f0)) c= F .: (X \ f0) ; ::_thesis: contradiction
then consider y being set such that
A29: y in rng (h | (X \ f0)) and
A30: not y in F .: (X \ f0) by TARSKI:def_3;
consider x1 being set such that
A31: x1 in dom (h | (X \ f0)) and
A32: (h | (X \ f0)) . x1 = y by A29, FUNCT_1:def_3;
A33: h . x1 = y by A31, A32, FUNCT_1:47;
x1 in (dom h) /\ (X \ f0) by A31, RELAT_1:61;
then A34: x1 in X \ f0 by XBOOLE_0:def_4;
A35: F .: (X \ (X \ f0)) = (F .: X) \ (F .: (X \ f0)) by A2, FUNCT_1:64;
rngF = F .: X by A1, RELAT_1:113;
then y in (F .: X) \ (F .: (X \ f0)) by A29, A30, XBOOLE_0:def_5;
then consider x2 being set such that
A36: x2 in dom F and
A37: x2 in X \ (X \ f0) and
A38: y = F . x2 by A35, FUNCT_1:def_6;
y in rng F by A36, A38, FUNCT_1:def_3;
then A39: X = dom h by FUNCT_2:def_1;
X \ (X \ f0) = X /\ (f " {0}) by XBOOLE_1:48;
then x2 in f " {0} by A37, XBOOLE_0:def_4;
then A40: h . x2 = y by A26, A38;
not x2 in X \ f0 by A37, XBOOLE_0:def_5;
hence contradiction by A25, A36, A40, A33, A39, A34, FUNCT_1:def_4; ::_thesis: verum
end;
hence contradiction by A19, A24, A25, A26; ::_thesis: verum
end;
A41: X,rngF are_equipotent by A1, A2, WELLORD2:def_4;
then A42: card rngF = card X by CARD_1:5;
card rngF = card X by A41, CARD_1:5;
then A43: ( rngF = {} implies X is empty ) ;
A44: F is Function of X,rngF by A1, FUNCT_2:1;
{ h where h is Function of X,rngF : ( h is one-to-one & rng (h | (X \ f0)) c= F .: (X \ f0) & ( for x being set st x in f0 holds
h . x = F . x ) ) } c= Intersection (Fy,f,0)
proof
assume not { h where h is Function of X,rngF : ( h is one-to-one & rng (h | (X \ f0)) c= F .: (X \ f0) & ( for x being set st x in f0 holds
h . x = F . x ) ) } c= Intersection (Fy,f,0) ; ::_thesis: contradiction
then consider z being set such that
A45: z in { h where h is Function of X,rngF : ( h is one-to-one & rng (h | (X \ f0)) c= F .: (X \ f0) & ( for x being set st x in f0 holds
h . x = F . x ) ) } and
A46: not z in Intersection (Fy,f,0) by TARSKI:def_3;
consider h being Function of X,(rng F) such that
A47: h = z and
A48: h is one-to-one and
rng (h | (X \ f0)) c= F .: (X \ f0) and
A49: for x being set st x in f0 holds
h . x = F . x by A45;
consider x being set such that
A50: x in f " {0} by A16, CARD_1:27, XBOOLE_0:def_1;
x in X by A50;
then x in dom Fy9 by FUNCT_2:def_1;
then A51: Fy9 . x in rng Fy9 by FUNCT_1:def_3;
A52: Fy9 . x = H1(x) by A6, A50;
h . x = F . x by A49, A50;
then h in Fy9 . x by A48, A52;
then h in union (rng Fy9) by A51, TARSKI:def_4;
then consider y being set such that
A53: y in dom f and
A54: f . y = 0 and
A55: not h in Fy . y by A46, A47, Def2;
f . y in {0} by A54, TARSKI:def_1;
then y in f " {0} by A53, FUNCT_1:def_7;
then h . y = F . y by A49;
then h in H1(y) by A48;
hence contradiction by A6, A53, A55; ::_thesis: verum
end;
then { h where h is Function of X,rngF : ( h is one-to-one & rng (h | (X \ f0)) c= F .: (X \ f0) & ( for x being set st x in f0 holds
h . x = F . x ) ) } = Intersection (Fy,f,0) by A17, XBOOLE_0:def_10;
then card (Intersection (Fy,f,0)) = ((card X) -' (n + 1)) ! by A2, A16, A43, A44, A42, Th61;
hence card (Intersection (Fy,f9,0)) = XF . n by A9, A10, A12, A15, A14; ::_thesis: verum
end;
A56: X,rngF are_equipotent by A1, A2, WELLORD2:def_4;
then card rngF = card X by CARD_1:5;
then A57: ( ((card rngF) -' (card X)) ! = 1 & card { f where f is Function of X,rngF : f is one-to-one } = ((card rngF) !) / (((card rngF) -' (card X)) !) ) by Th7, NEWTON:12, XREAL_1:232;
then reconsider One = { f where f is Function of X,(rng F) : f is one-to-one } as finite set ;
set S = { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } ;
( dom XF = card X & dom Fy = X ) by FUNCT_2:def_1;
then consider F9 being XFinSequence of such that
A58: dom F9 = card X and
A59: card (union (rng Fy)) = Sum F9 and
A60: for n being Nat st n in dom F9 holds
F9 . n = (((- 1) |^ n) * (XF . n)) * ((card X) choose (n + 1)) by A11, Th58;
A61: union (rng Fy9) c= One
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in union (rng Fy9) or x in One )
assume x in union (rng Fy9) ; ::_thesis: x in One
then consider Fyx being set such that
A62: x in Fyx and
A63: Fyx in rng Fy9 by TARSKI:def_4;
consider x1 being set such that
A64: ( x1 in dom Fy9 & Fy . x1 = Fyx ) by A63, FUNCT_1:def_3;
x in H1(x1) by A6, A62, A64;
then ex h being Function of X,(rng F) st
( h = x & h is one-to-one & h . x1 = F . x1 ) ;
hence x in One ; ::_thesis: verum
end;
reconsider u = union (rng Fy) as finite set ;
A65: card (One \ u) = (card One) - (card u) by A61, CARD_2:44;
take F9 ; ::_thesis: ( dom F9 = card X & ((card X) !) - (Sum F9) = card { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } & ( for n being Nat st n in dom F9 holds
F9 . n = (((- 1) |^ n) * ((card X) !)) / ((n + 1) !) ) )
thus dom F9 = card X by A58; ::_thesis: ( ((card X) !) - (Sum F9) = card { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } & ( for n being Nat st n in dom F9 holds
F9 . n = (((- 1) |^ n) * ((card X) !)) / ((n + 1) !) ) )
A66: One \ (union (rng Fy)) c= { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in One \ (union (rng Fy)) or x in { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } )
assume A67: x in One \ (union (rng Fy)) ; ::_thesis: x in { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) }
x in One by A67;
then consider f being Function of X,(rng F) such that
A68: f = x and
A69: f is one-to-one ;
assume not x in { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } ; ::_thesis: contradiction
then consider x being set such that
A70: x in X and
A71: f . x = F . x by A68, A69;
x in dom Fy by A70, FUNCT_2:def_1;
then Fy . x in rng Fy by FUNCT_1:def_3;
then A72: H1(x) in rng Fy by A6, A70;
f in H1(x) by A69, A71;
then f in union (rng Fy) by A72, TARSKI:def_4;
hence contradiction by A67, A68, XBOOLE_0:def_5; ::_thesis: verum
end;
A73: { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } c= One \ (union (rng Fy))
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } or x in One \ (union (rng Fy)) )
assume x in { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } ; ::_thesis: x in One \ (union (rng Fy))
then consider f being Function of X,(rng F) such that
A74: x = f and
A75: f is one-to-one and
A76: for x being set st x in X holds
f . x <> F . x ;
assume A77: not x in One \ (union (rng Fy)) ; ::_thesis: contradiction
f in One by A75;
then f in union (rng Fy) by A74, A77, XBOOLE_0:def_5;
then consider Fyy being set such that
A78: f in Fyy and
A79: Fyy in rng Fy by TARSKI:def_4;
consider y being set such that
A80: y in dom Fy and
A81: Fy . y = Fyy by A79, FUNCT_1:def_3;
y in X by A80, FUNCT_2:def_1;
then f in H1(y) by A6, A78, A81;
then A82: ex g being Function of X,(rng F) st
( f = g & g is one-to-one & g . y = F . y ) ;
y in X by A80, FUNCT_2:def_1;
hence contradiction by A76, A82; ::_thesis: verum
end;
card One = (card X) ! by A56, A57, CARD_1:5;
hence card { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } = ((card X) !) - (Sum F9) by A59, A66, A73, A65, XBOOLE_0:def_10; ::_thesis: for n being Nat st n in dom F9 holds
F9 . n = (((- 1) |^ n) * ((card X) !)) / ((n + 1) !)
let n be Nat; ::_thesis: ( n in dom F9 implies F9 . n = (((- 1) |^ n) * ((card X) !)) / ((n + 1) !) )
assume A83: n in dom F9 ; ::_thesis: F9 . n = (((- 1) |^ n) * ((card X) !)) / ((n + 1) !)
n < card X by A58, A83, NAT_1:44;
then A84: n + 1 <= card X by NAT_1:13;
then reconsider c = (card X) - (n + 1) as Element of NAT by NAT_1:21;
A85: (card X) choose (n + 1) = ((card X) !) / ((c !) * ((n + 1) !)) by A84, NEWTON:def_3;
A86: c ! > 0 by NEWTON:17;
XF . n = c ! by A9, A58, A83;
then A87: (XF . n) * ((card X) choose (n + 1)) = ((c !) * ((card X) !)) / ((c !) * ((n + 1) !)) by A85, XCMPLX_1:74
.= ((card X) !) * ((c !) / ((c !) * ((n + 1) !))) by XCMPLX_1:74
.= ((card X) !) * (((c !) / (c !)) / ((n + 1) !)) by XCMPLX_1:78
.= ((card X) !) * (1 / ((n + 1) !)) by A86, XCMPLX_1:60
.= (((card X) !) * 1) / ((n + 1) !) by XCMPLX_1:74 ;
F9 . n = (((- 1) |^ n) * (XF . n)) * ((card X) choose (n + 1)) by A60, A83
.= ((- 1) |^ n) * (((card X) !) / ((n + 1) !)) by A87 ;
hence F9 . n = (((- 1) |^ n) * ((card X) !)) / ((n + 1) !) by XCMPLX_1:74; ::_thesis: verum
end;
theorem Th62: :: CARD_FIN:62
for X being finite set
for F being Function st dom F = X & F is one-to-one holds
ex XF being XFinSequence of st
( Sum XF = card { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } & dom XF = (card X) + 1 & ( for n being Nat st n in dom XF holds
XF . n = (((- 1) |^ n) * ((card X) !)) / (n !) ) )
proof
let X be finite set ; ::_thesis: for F being Function st dom F = X & F is one-to-one holds
ex XF being XFinSequence of st
( Sum XF = card { h where h is Function of X,(rng F) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F . x ) ) } & dom XF = (card X) + 1 & ( for n being Nat st n in dom XF holds
XF . n = (((- 1) |^ n) * ((card X) !)) / (n !) ) )
let F9 be Function; ::_thesis: ( dom F9 = X & F9 is one-to-one implies ex XF being XFinSequence of st
( Sum XF = card { h where h is Function of X,(rng F9) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F9 . x ) ) } & dom XF = (card X) + 1 & ( for n being Nat st n in dom XF holds
XF . n = (((- 1) |^ n) * ((card X) !)) / (n !) ) ) )
assume that
A1: dom F9 = X and
A2: F9 is one-to-one ; ::_thesis: ex XF being XFinSequence of st
( Sum XF = card { h where h is Function of X,(rng F9) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F9 . x ) ) } & dom XF = (card X) + 1 & ( for n being Nat st n in dom XF holds
XF . n = (((- 1) |^ n) * ((card X) !)) / (n !) ) )
X, rng F9 are_equipotent by A1, A2, WELLORD2:def_4;
then card X = card (rng F9) by CARD_1:5;
then reconsider rngF = rng F9 as finite set ;
reconsider F = F9 as Function of X,rngF by A1, FUNCT_2:1;
set S = { h where h is Function of X,(rng F9) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F9 . x ) ) } ;
rng F9 = rng F ;
then consider Xf being XFinSequence of such that
A3: dom Xf = card X and
A4: ((card X) !) - (Sum Xf) = card { h where h is Function of X,(rng F9) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F9 . x ) ) } and
A5: for n being Nat st n in dom Xf holds
Xf . n = (((- 1) |^ n) * ((card X) !)) / ((n + 1) !) by A1, A2, Lm4;
reconsider c = (card X) ! as Element of INT by INT_1:def_2;
A6: len <%c%> = 1 by AFINSQ_1:33;
set F1 = (- 1) (#) Xf;
A7: dom ((- 1) (#) Xf) = card X by A3, VALUED_1:def_5;
reconsider F1 = (- 1) (#) Xf as XFinSequence of ;
set XF = <%c%> ^ F1;
take <%c%> ^ F1 ; ::_thesis: ( Sum (<%c%> ^ F1) = card { h where h is Function of X,(rng F9) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F9 . x ) ) } & dom (<%c%> ^ F1) = (card X) + 1 & ( for n being Nat st n in dom (<%c%> ^ F1) holds
(<%c%> ^ F1) . n = (((- 1) |^ n) * ((card X) !)) / (n !) ) )
(- 1) * (Sum Xf) = Sum F1 by AFINSQ_2:64;
then c - (Sum Xf) = c + (Sum F1)
.= addint . (c,(Sum F1)) by BINOP_2:def_20
.= addint . ((addint "**" <%c%>),(Sum F1)) by AFINSQ_2:37
.= addint . ((addint "**" <%c%>),(addint "**" F1)) by AFINSQ_2:50
.= addint "**" (<%c%> ^ F1) by AFINSQ_2:42
.= Sum (<%c%> ^ F1) by AFINSQ_2:50 ;
hence Sum (<%c%> ^ F1) = card { h where h is Function of X,(rng F9) : ( h is one-to-one & ( for x being set st x in X holds
h . x <> F9 . x ) ) } by A4; ::_thesis: ( dom (<%c%> ^ F1) = (card X) + 1 & ( for n being Nat st n in dom (<%c%> ^ F1) holds
(<%c%> ^ F1) . n = (((- 1) |^ n) * ((card X) !)) / (n !) ) )
len F1 = card X by A3, VALUED_1:def_5;
hence A8: dom (<%c%> ^ F1) = (card X) + 1 by A6, AFINSQ_1:def_3; ::_thesis: for n being Nat st n in dom (<%c%> ^ F1) holds
(<%c%> ^ F1) . n = (((- 1) |^ n) * ((card X) !)) / (n !)
let n be Nat; ::_thesis: ( n in dom (<%c%> ^ F1) implies (<%c%> ^ F1) . n = (((- 1) |^ n) * ((card X) !)) / (n !) )
assume A9: n in dom (<%c%> ^ F1) ; ::_thesis: (<%c%> ^ F1) . n = (((- 1) |^ n) * ((card X) !)) / (n !)
percases ( n = 0 or n > 0 ) ;
supposeA10: n = 0 ; ::_thesis: (<%c%> ^ F1) . n = (((- 1) |^ n) * ((card X) !)) / (n !)
then (- 1) |^ n = 1 by NEWTON:4;
hence (<%c%> ^ F1) . n = (((- 1) |^ n) * ((card X) !)) / (n !) by A10, AFINSQ_1:35, NEWTON:12; ::_thesis: verum
end;
suppose n > 0 ; ::_thesis: (<%c%> ^ F1) . n = (((- 1) |^ n) * ((card X) !)) / (n !)
then reconsider n1 = n - 1 as Element of NAT by NAT_1:20;
n1 + 1 = n ;
then A11: (- 1) * ((- 1) |^ n1) = (- 1) |^ n by NEWTON:6;
n < (card X) + 1 by A8, A9, NAT_1:44;
then n1 + 1 <= card X by NAT_1:13;
then n1 < len F1 by A7, NAT_1:13;
then A12: n1 in dom F1 by NAT_1:44;
len <%c%> = 1 by AFINSQ_1:33;
then (<%c%> ^ F1) . (n1 + 1) = F1 . n1 by A12, AFINSQ_1:def_3;
then A13: (<%c%> ^ F1) . (n1 + 1) = (- 1) * (Xf . n1) by VALUED_1:6;
Xf . n1 = (((- 1) |^ n1) * ((card X) !)) / ((n1 + 1) !) by A3, A5, A7, A12;
then (<%c%> ^ F1) . n = ((- 1) * (((- 1) |^ n1) * ((card X) !))) / (n !) by A13, XCMPLX_1:74;
hence (<%c%> ^ F1) . n = (((- 1) |^ n) * ((card X) !)) / (n !) by A11; ::_thesis: verum
end;
end;
end;
theorem :: CARD_FIN:63
for X being finite set ex XF being XFinSequence of st
( Sum XF = card { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> x ) ) } & dom XF = (card X) + 1 & ( for n being Nat st n in dom XF holds
XF . n = (((- 1) |^ n) * ((card X) !)) / (n !) ) )
proof
let X be finite set ; ::_thesis: ex XF being XFinSequence of st
( Sum XF = card { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> x ) ) } & dom XF = (card X) + 1 & ( for n being Nat st n in dom XF holds
XF . n = (((- 1) |^ n) * ((card X) !)) / (n !) ) )
set S1 = { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> (id X) . x ) ) } ;
set S2 = { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> x ) ) } ;
A1: { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> x ) ) } c= { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> (id X) . x ) ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> x ) ) } or x in { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> (id X) . x ) ) } )
assume x in { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> x ) ) } ; ::_thesis: x in { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> (id X) . x ) ) }
then consider h being Function of X,X such that
A2: ( h = x & h is one-to-one ) and
A3: for y being set st y in X holds
h . y <> y ;
now__::_thesis:_for_y_being_set_st_y_in_X_holds_
(id_X)_._y_<>_h_._y
let y be set ; ::_thesis: ( y in X implies (id X) . y <> h . y )
assume A4: y in X ; ::_thesis: (id X) . y <> h . y
(id X) . y = y by A4, FUNCT_1:17;
hence (id X) . y <> h . y by A3, A4; ::_thesis: verum
end;
hence x in { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> (id X) . x ) ) } by A2; ::_thesis: verum
end;
A5: ( dom (id X) = X & rng (id X) = X ) ;
{ h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> (id X) . x ) ) } c= { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> x ) ) }
proof
let x be set ; :: according to TARSKI:def_3 ::_thesis: ( not x in { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> (id X) . x ) ) } or x in { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> x ) ) } )
assume x in { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> (id X) . x ) ) } ; ::_thesis: x in { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> x ) ) }
then consider h being Function of X,X such that
A6: ( h = x & h is one-to-one ) and
A7: for y being set st y in X holds
h . y <> (id X) . y ;
now__::_thesis:_for_y_being_set_st_y_in_X_holds_
h_._y_<>_y
let y be set ; ::_thesis: ( y in X implies h . y <> y )
assume A8: y in X ; ::_thesis: h . y <> y
(id X) . y = y by A8, FUNCT_1:17;
hence h . y <> y by A7, A8; ::_thesis: verum
end;
hence x in { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> x ) ) } by A6; ::_thesis: verum
end;
then { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> (id X) . x ) ) } = { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> x ) ) } by A1, XBOOLE_0:def_10;
hence ex XF being XFinSequence of st
( Sum XF = card { h where h is Function of X,X : ( h is one-to-one & ( for x being set st x in X holds
h . x <> x ) ) } & dom XF = (card X) + 1 & ( for n being Nat st n in dom XF holds
XF . n = (((- 1) |^ n) * ((card X) !)) / (n !) ) ) by A5, Th62; ::_thesis: verum
end;