:: The Contraction Lemma
:: by Grzegorz Bancerek
::
:: Copyright (c) 1990-2021 Association of Mizar Users

definition
let E be non empty set ;
let A be Ordinal;
deffunc H1( Sequence) -> set = { d where d is Element of E : for d1 being Element of E st d1 in d holds
ex C being Ordinal st
( C in dom $1 & d1 in union {($1 . C)} )
}
;
func Collapse (E,A) -> set means :Def1: :: ZF_COLLA:def 1
ex L being Sequence st
( it = { d where d is Element of E : for d1 being Element of E st d1 in d holds
ex B being Ordinal st
( B in dom L & d1 in union {(L . B)} )
}
& dom L = A & ( for B being Ordinal st B in A holds
L . B = { d1 where d1 is Element of E : for d being Element of E st d in d1 holds
ex C being Ordinal st
( C in dom (L | B) & d in union {((L | B) . C)} )
}
) );
existence
ex b1 being set ex L being Sequence st
( b1 = { d where d is Element of E : for d1 being Element of E st d1 in d holds
ex B being Ordinal st
( B in dom L & d1 in union {(L . B)} )
}
& dom L = A & ( for B being Ordinal st B in A holds
L . B = { d1 where d1 is Element of E : for d being Element of E st d in d1 holds
ex C being Ordinal st
( C in dom (L | B) & d in union {((L | B) . C)} )
}
) )
proof end;
uniqueness
for b1, b2 being set st ex L being Sequence st
( b1 = { d where d is Element of E : for d1 being Element of E st d1 in d holds
ex B being Ordinal st
( B in dom L & d1 in union {(L . B)} )
}
& dom L = A & ( for B being Ordinal st B in A holds
L . B = { d1 where d1 is Element of E : for d being Element of E st d in d1 holds
ex C being Ordinal st
( C in dom (L | B) & d in union {((L | B) . C)} )
}
) ) & ex L being Sequence st
( b2 = { d where d is Element of E : for d1 being Element of E st d1 in d holds
ex B being Ordinal st
( B in dom L & d1 in union {(L . B)} )
}
& dom L = A & ( for B being Ordinal st B in A holds
L . B = { d1 where d1 is Element of E : for d being Element of E st d in d1 holds
ex C being Ordinal st
( C in dom (L | B) & d in union {((L | B) . C)} )
}
) ) holds
b1 = b2
proof end;
end;

:: deftheorem Def1 defines Collapse ZF_COLLA:def 1 :
for E being non empty set
for A being Ordinal
for b3 being set holds
( b3 = Collapse (E,A) iff ex L being Sequence st
( b3 = { d where d is Element of E : for d1 being Element of E st d1 in d holds
ex B being Ordinal st
( B in dom L & d1 in union {(L . B)} )
}
& dom L = A & ( for B being Ordinal st B in A holds
L . B = { d1 where d1 is Element of E : for d being Element of E st d in d1 holds
ex C being Ordinal st
( C in dom (L | B) & d in union {((L | B) . C)} )
}
) ) );

theorem Th1: :: ZF_COLLA:1
for E being non empty set
for A being Ordinal holds Collapse (E,A) = { d where d is Element of E : for d1 being Element of E st d1 in d holds
ex B being Ordinal st
( B in A & d1 in Collapse (E,B) )
}
proof end;

theorem :: ZF_COLLA:2
for E being non empty set
for d being Element of E holds
( ( for d1 being Element of E holds not d1 in d ) iff d in Collapse (E,{}) )
proof end;

theorem :: ZF_COLLA:3
for E being non empty set
for A being Ordinal
for d being Element of E holds
( d /\ E c= Collapse (E,A) iff d in Collapse (E,(succ A)) )
proof end;

theorem Th4: :: ZF_COLLA:4
for E being non empty set
for A, B being Ordinal st A c= B holds
Collapse (E,A) c= Collapse (E,B)
proof end;

theorem Th5: :: ZF_COLLA:5
for E being non empty set
for d being Element of E ex A being Ordinal st d in Collapse (E,A)
proof end;

theorem Th6: :: ZF_COLLA:6
for E being non empty set
for A being Ordinal
for d, d9 being Element of E st d9 in d & d in Collapse (E,A) holds
( d9 in Collapse (E,A) & ex B being Ordinal st
( B in A & d9 in Collapse (E,B) ) )
proof end;

theorem Th7: :: ZF_COLLA:7
for E being non empty set
for A being Ordinal holds Collapse (E,A) c= E
proof end;

theorem Th8: :: ZF_COLLA:8
for E being non empty set ex A being Ordinal st E = Collapse (E,A)
proof end;

theorem Th9: :: ZF_COLLA:9
for E being non empty set ex f being Function st
( dom f = E & ( for d being Element of E holds f . d = f .: d ) )
proof end;

:: Definition of epsilon-isomorphism of two sets
definition
let f be Function;
let X, Y be set ;
pred f is_epsilon-isomorphism_of X,Y means :: ZF_COLLA:def 2
( dom f = X & rng f = Y & f is one-to-one & ( for x, y being object st x in X & y in X holds
( ex Z being set st
( Z = y & x in Z ) iff ex Z being set st
( f . y = Z & f . x in Z ) ) ) );
end;

:: deftheorem defines is_epsilon-isomorphism_of ZF_COLLA:def 2 :
for f being Function
for X, Y being set holds
( f is_epsilon-isomorphism_of X,Y iff ( dom f = X & rng f = Y & f is one-to-one & ( for x, y being object st x in X & y in X holds
( ex Z being set st
( Z = y & x in Z ) iff ex Z being set st
( f . y = Z & f . x in Z ) ) ) ) );

definition
let X, Y be set ;
pred X,Y are_epsilon-isomorphic means :: ZF_COLLA:def 3
ex f being Function st f is_epsilon-isomorphism_of X,Y;
end;

:: deftheorem defines are_epsilon-isomorphic ZF_COLLA:def 3 :
for X, Y being set holds
( X,Y are_epsilon-isomorphic iff ex f being Function st f is_epsilon-isomorphism_of X,Y );

theorem Th10: :: ZF_COLLA:10
for E being non empty set
for f being Function st dom f = E & ( for d being Element of E holds f . d = f .: d ) holds
rng f is epsilon-transitive
proof end;

theorem Th11: :: ZF_COLLA:11
for E being non empty set st E |= the_axiom_of_extensionality holds
for u, v being Element of E st ( for w being Element of E holds
( w in u iff w in v ) ) holds
u = v
proof end;

:: Contraction Lemma
theorem :: ZF_COLLA:12
for E being non empty set st E |= the_axiom_of_extensionality holds
ex X being set st
( X is epsilon-transitive & E,X are_epsilon-isomorphic )
proof end;