Theorem funfni 4513

Description: Inference to convert a function and domain antecedent.

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 4510	. . 3 |- (F Fn A -> Fun F)
2	1	adantr 425	. 2 |- ((F Fn A /\ B e. A) -> Fun F)
3		fndm 4512	. . . 4 |- (F Fn A -> dom F = A)
4	3	eleq2d 1964	. . 3 |- (F Fn A -> (B e. dom F <-> B e. A))
5	4	biimpar 461	. 2 |- ((F Fn A /\ B e. A) -> B e. dom F)
6		funfni.1	. 2 |- ((Fun F /\ B e. dom F) -> ph)
7	2, 5, 6	syl11anc 524	1 |- ((F Fn A /\ B e. A) -> ph)

Syntax hints:   -> wi 3   /\ wa 240   e. wcel 1300  dom cdm 3986  Fun wfun 3992   Fn wfn 3993

This theorem is referenced by:  fneu 4517  fvco2 4737  fvco2OLD 4738  fnopfv 4784  fnfvelrn 4786  isomin 4876  isofrlem 4878  elpreima 10161  fipreima 10175  fipreimaOLD 15756

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 1305  ax-17 1317  ax-4 1319  ax-5o 1321  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-an 242  df-ex 1327  df-cleq 1877  df-clel 1880  df-fn 4009

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