:: Mostowski's Fundamental Operations - Part II
:: by Grzegorz Bancerek and Andrzej Kondracki
::
:: Received February 15, 1991
:: Copyright (c) 1991-2016 Association of Mizar Users

environ

vocabularies HIDDEN, NUMBERS, ZF_LANG, XBOOLE_0, SUBSET_1, FUNCT_1, ARYTM_3, CARD_1, ZF_MODEL, TARSKI, ORDINAL1, BVFUNC_2, XBOOLEAN, ZFMISC_1, CLASSES2, ZF_REFLE, CARD_3, RELAT_1, ORDINAL2, ZF_LANG1, ZFMODEL1, XXREAL_0, REALSET1, ZF_FUND1, FUNCT_2, ORDINAL4, NAT_1, ZF_FUND2;
notations HIDDEN, TARSKI, XBOOLE_0, ENUMSET1, ZFMISC_1, SUBSET_1, CARD_1, ORDINAL1, XCMPLX_0, NAT_1, RELAT_1, FUNCT_1, RELSET_1, PARTFUN1, FUNCT_2, ZF_LANG, ZF_MODEL, ZFMODEL1, ORDINAL2, NUMBERS, CARD_3, ZF_LANG1, CLASSES2, ORDINAL4, ZF_REFLE, ZF_FUND1, XXREAL_0;
definitions TARSKI, ORDINAL1, XBOOLE_0;
theorems TARSKI, NAT_1, ENUMSET1, FUNCT_1, FUNCT_2, ORDINAL1, ORDINAL2, CARD_1, ZF_MODEL, ZFMODEL1, ZFMODEL2, ZF_LANG1, CARD_3, ZF_REFLE, ZFREFLE1, CLASSES2, ZFMISC_1, ZF_FUND1, GRFUNC_1, RELAT_1, CLASSES1, RELSET_1, XBOOLE_0, XBOOLE_1, XREAL_1, FUNCT_7;
schemes ZF_REFLE, NAT_1;
registrations ORDINAL1, RELSET_1, FINSET_1, XREAL_0, CARD_1, CLASSES2, ZF_LANG, ORDINAL4, FUNCT_1, ZF_FUND1, ZF_LANG1;
constructors HIDDEN, PARTFUN1, WELLORD2, DOMAIN_1, XXREAL_0, XREAL_0, NAT_1, CLASSES1, ZFMODEL1, ZF_LANG1, ZF_REFLE, ZF_FUND1, ORDINAL4, RELSET_1, ZF_MODEL, NUMBERS;
requirements HIDDEN, NUMERALS, REAL, SUBSET, BOOLE, ARITHM;
equalities ORDINAL1, XBOOLE_0;
expansions TARSKI, ORDINAL1, XBOOLE_0;


definition
let H be ZF-formula;
let M be non empty set ;
let v be Function of VAR,M;
func Section (H,v) -> Subset of M equals :Def1: :: ZF_FUND2:def 1
 { m where m is Element of M : M,v / ((x. 0),m) |= H } if x. 0 in Free H
otherwise {} ;
coherence
( ( x. 0 in Free H implies { m where m is Element of M : M,v / ((x. 0),m) |= H } is Subset of M ) & ( not x. 0 in Free H implies {} is Subset of M ) )
proof end;
consistency
for b1 being Subset of M holds verum ;
end;

:: deftheorem Def1 defines Section ZF_FUND2:def 1 :
for H being ZF-formula
for M being non empty set
for v being Function of VAR,M holds
( ( x. 0 in Free H implies Section (H,v) = { m where m is Element of M : M,v / ((x. 0),m) |= H } ) & ( not x. 0 in Free H implies Section (H,v) = {} ) );

definition
let M be non empty set ;
attr M is predicatively_closed means :: ZF_FUND2:def 2
 for H being ZF-formula
for E being non empty set
for f being Function of VAR,E st E in M holds
Section (H,f) in M;
end;

:: deftheorem defines predicatively_closed ZF_FUND2:def 2 :
for M being non empty set holds
( M is predicatively_closed iff for H being ZF-formula
for E being non empty set
for f being Function of VAR,E st E in M holds
Section (H,f) in M );

theorem Th1: :: ZF_FUND2:1
for E being non empty set
for e being Element of E
for f being Function of VAR,E st E is epsilon-transitive holds
Section ((All ((x. 2),(((x. 2) 'in' (x. 0)) => ((x. 2) 'in' (x. 1))))),(f / ((x. 1),e))) = E /\ (bool e)
proof end;

Lm1: for g being Function
for u1 being set st u1 in Union g holds
ex u2 being set st
( u2 in dom g & u1 in g . u2 )
proof end;

theorem Th2: :: ZF_FUND2:2
for W being Universe
for L being DOMAIN-Sequence of W st ( for a, b being Ordinal of W st a in b holds
L . a c= L . b ) & ( for a being Ordinal of W holds
( L . a in Union L & L . a is epsilon-transitive ) ) & Union L is predicatively_closed holds
Union L |= the_axiom_of_power_sets
proof end;

Lm2: for H being ZF-formula
for M being non empty set
for v being Function of VAR,M
for x being Variable st not x in variables_in H & {(x. 0),(x. 1),(x. 2)} misses Free H & M,v |= All ((x. 3),(Ex ((x. 0),(All ((x. 4),(H <=> ((x. 4) '=' (x. 0)))))))) holds
( {(x. 0),(x. 1),(x. 2)} misses Free (H / ((x. 0),x)) & M,v |= All ((x. 3),(Ex ((x. 0),(All ((x. 4),((H / ((x. 0),x)) <=> ((x. 4) '=' (x. 0)))))))) & def_func' (H,v) = def_func' ((H / ((x. 0),x)),v) )
proof end;

Lm3: for H being ZF-formula
for M being non empty set
for v being Function of VAR,M st M,v |= All ((x. 3),(Ex ((x. 0),(All ((x. 4),(H <=> ((x. 4) '=' (x. 0)))))))) & not x. 4 in Free H holds
for m being Element of M holds (def_func' (H,v)) .: m = {}
proof end;

Lm4: for H being ZF-formula
for x, y being Variable st not y in variables_in H & x <> x. 0 & y <> x. 0 & y <> x. 4 holds
( x. 4 in Free H iff x. 0 in Free (Ex ((x. 3),(((x. 3) 'in' x) '&' ((H / ((x. 0),y)) / ((x. 4),(x. 0)))))) )
proof end;

theorem Th3: :: ZF_FUND2:3
for W being Universe
for L being DOMAIN-Sequence of W st omega in W & ( for a, b being Ordinal of W st a in b holds
L . a c= L . b ) & ( for a being Ordinal of W st a <> {} & a is limit_ordinal holds
L . a = Union (L | a) ) & ( for a being Ordinal of W holds
( L . a in Union L & L . a is epsilon-transitive ) ) & Union L is predicatively_closed holds
for H being ZF-formula st {(x. 0),(x. 1),(x. 2)} misses Free H holds
Union L |= the_axiom_of_substitution_for H
proof end;

Lm5: for H being ZF-formula
for M being non empty set
for m being Element of M
for v being Function of VAR,M
for i being Element of NAT st x. i in Free H holds
{[i,m]} \/ ((v * decode) | ((code (Free H)) \ {i})) = ((v / ((x. i),m)) * decode) | (code (Free H))
proof end;

theorem Th4: :: ZF_FUND2:4
for H being ZF-formula
for M being non empty set
for v being Function of VAR,M holds Section (H,v) = { m where m is Element of M : {[{},m]} \/ ((v * decode) | ((code (Free H)) \ {{}})) in Diagram (H,M) }
proof end;

theorem Th5: :: ZF_FUND2:5
for W being Universe
for Y being Subclass of W st Y is closed_wrt_A1-A7 & Y is epsilon-transitive holds
Y is predicatively_closed
proof end;

theorem :: ZF_FUND2:6
for W being Universe
for L being DOMAIN-Sequence of W st omega in W & ( for a, b being Ordinal of W st a in b holds
L . a c= L . b ) & ( for a being Ordinal of W st a <> {} & a is limit_ordinal holds
L . a = Union (L | a) ) & ( for a being Ordinal of W holds
( L . a in Union L & L . a is epsilon-transitive ) ) & Union L is closed_wrt_A1-A7 holds
Union L is being_a_model_of_ZF
proof end;