Theorem fabexg

Description: Existence of a set of functions. (Contributed by Paul Chapman, 25-Feb-2008) (Proof shortened by AV, 9-Jun-2025)

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

Proof

Step	Hyp	Ref	Expression
1		fabexg.1	|- F = { x | ( x : A --> B /\ ph ) }
2		elex	|- ( A e. C -> A e. _V )
3		elex	|- ( B e. D -> B e. _V )
4		simprl	|- ( ( ( A e. _V /\ B e. _V ) /\ ( x : A --> B /\ ph ) ) -> x : A --> B )
5		simpl	|- ( ( A e. _V /\ B e. _V ) -> A e. _V )
6		simpr	|- ( ( A e. _V /\ B e. _V ) -> B e. _V )
7	4 5 6	fabexd	|- ( ( A e. _V /\ B e. _V ) -> { x | ( x : A --> B /\ ph ) } e. _V )
8	1 7	eqeltrid	|- ( ( A e. _V /\ B e. _V ) -> F e. _V )
9	2 3 8	syl2an	|- ( ( A e. C /\ B e. D ) -> F e. _V )