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) (Proof shortened by AV, 9-Jun-2025)

	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		elex	|- ( A e. C -> A e. _V )
3		elex	|- ( B e. D -> B e. _V )
4		f1of	|- ( f : A -1-1-onto-> B -> f : A --> B )
5	4	ad2antrl	|- ( ( ( A e. _V /\ B e. _V ) /\ ( f : A -1-1-onto-> B /\ ph ) ) -> f : A --> B )
6		simpl	|- ( ( A e. _V /\ B e. _V ) -> A e. _V )
7		simpr	|- ( ( A e. _V /\ B e. _V ) -> B e. _V )
8	5 6 7	fabexd	|- ( ( A e. _V /\ B e. _V ) -> { f | ( f : A -1-1-onto-> B /\ ph ) } e. _V )
9	1 8	eqeltrid	|- ( ( A e. _V /\ B e. _V ) -> F e. _V )
10	2 3 9	syl2an	|- ( ( A e. C /\ B e. D ) -> F e. _V )