Theorem funfni 3584

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		funfni.1	. 2 |- ((Fun F /\ B e. dom F) -> ph)
2		fnfun 3581	. . 3 |- (F Fn A -> Fun F)
3	2	adantr 389	. 2 |- ((F Fn A /\ B e. A) -> Fun F)
4		fndm 3583	. . . 4 |- (F Fn A -> dom F = A)
5	4	eleq2d 1539	. . 3 |- (F Fn A -> (B e. dom F <-> B e. A))
6	5	biimpar 417	. 2 |- ((F Fn A /\ B e. A) -> B e. dom F)
7	1, 3, 6	sylanc 471	1 |- ((F Fn A /\ B e. A) -> ph)

Syntax hints:   -> wi 3   /\ wa 223   e. wcel 957  dom cdm 3166  Fun wfun 3172   Fn wfn 3173

This theorem is referenced by:  fvco2 3770  fnopfv 3806  fnfvelrn 3808  isomin 3894  isofrlem 3896

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-gen 962  ax-17 970  ax-4 972  ax-5o 974  ax-ext 1458

This theorem depends on definitions:  df-bi 147  df-an 225  df-ex 980  df-cleq 1468  df-clel 1471  df-fn 3189

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