Theorem funfni

Description: Inference to convert a function and domain antecedent. (Contributed by NM, 22-Apr-2004)

	Ref	Expression
Hypothesis	funfni.1	|- ( ( Fun F /\ B e. dom F ) -> ph )
Assertion	funfni	|- ( ( F Fn A /\ B e. A ) -> ph )

Proof

Step	Hyp	Ref	Expression
1		funfni.1	|- ( ( Fun F /\ B e. dom F ) -> ph )
2		fnfun	|- ( F Fn A -> Fun F )
3		fndm	|- ( F Fn A -> dom F = A )
4	3	eleq2d	|- ( F Fn A -> ( B e. dom F <-> B e. A ) )
5	4	biimpar	|- ( ( F Fn A /\ B e. A ) -> B e. dom F )
6	2 5 1	syl2an2r	|- ( ( F Fn A /\ B e. A ) -> ph )