Theorem funfni 5991

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

Hypothesis

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

Assertion

Ref	Expression
funfni	|- ( ( F Fn A /\ B e. A ) -> ph )

Proof of Theorem funfni

Step	Hyp	Ref	Expression
1		fnfun 5988	. 2 |- ( F Fn A -> Fun F )
2		fndm 5990	. . . 4 |- ( F Fn A -> dom F = A )
3	2	eleq2d 2687	. . 3 |- ( F Fn A -> ( B e. dom F <-> B e. A ) )
4	3	biimpar 502	. 2 |- ( ( F Fn A /\ B e. A ) -> B e. dom F )
5		funfni.1	. 2 |- ( ( Fun F /\ B e. dom F ) -> ph )
6	1, 4, 5	syl2an2r 876	1 |- ( ( F Fn A /\ B e. A ) -> ph )

Syntax hints:   -> wi 4   /\ wa 384   e. wcel 1990   dom cdm 5114   Fun wfun 5882   Fn wfn 5883

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-an 386  df-ex 1705  df-cleq 2615  df-clel 2618  df-fn 5891

This theorem is referenced by:  fneu  5995  elpreima  6337  fnopfv  6351  fnfvelrn  6356  funressnfv  41208  fnafvelrn  41249  afvco2  41256