Theorem f1oabexg

Description: The class of all 1-1-onto functions mapping one set to another is a set. (Contributed by Paul Chapman, 25-Feb-2008)

	Ref	Expression
Hypothesis	f1oabexg.1	|- F = { f | ( f : A -1-1-onto-> B /\ ph ) }
Assertion	f1oabexg	|- ( ( A e. C /\ B e. D ) -> F e. _V )

Proof

Step	Hyp	Ref	Expression
1		f1oabexg.1	|- F = { f | ( f : A -1-1-onto-> B /\ ph ) }
2		f1of	|- ( f : A -1-1-onto-> B -> f : A --> B )
3	2	anim1i	|- ( ( f : A -1-1-onto-> B /\ ph ) -> ( f : A --> B /\ ph ) )
4	3	ss2abi	|- { f | ( f : A -1-1-onto-> B /\ ph ) } C_ { f | ( f : A --> B /\ ph ) }
5		eqid	|- { f | ( f : A --> B /\ ph ) } = { f | ( f : A --> B /\ ph ) }
6	5	fabexg	|- ( ( A e. C /\ B e. D ) -> { f | ( f : A --> B /\ ph ) } e. _V )
7		ssexg	|- ( ( { f | ( f : A -1-1-onto-> B /\ ph ) } C_ { f | ( f : A --> B /\ ph ) } /\ { f | ( f : A --> B /\ ph ) } e. _V ) -> { f | ( f : A -1-1-onto-> B /\ ph ) } e. _V )
8	4 6 7	sylancr	|- ( ( A e. C /\ B e. D ) -> { f | ( f : A -1-1-onto-> B /\ ph ) } e. _V )
9	1 8	eqeltrid	|- ( ( A e. C /\ B e. D ) -> F e. _V )