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Theorem funfni 5574
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
StepHypRef Expression
1 fnfun 5571 . . 3  |-  ( F  Fn  A  ->  Fun  F )
21adantr 453 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  Fun  F )
3 fndm 5573 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
43eleq2d 2509 . . 3  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
54biimpar 473 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  B  e.  dom  F
)
6 funfni.1 . 2  |-  ( ( Fun  F  /\  B  e.  dom  F )  ->  ph )
72, 5, 6syl2anc 644 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 360    e. wcel 1727   dom cdm 4907   Fun wfun 5477    Fn wfn 5478
This theorem is referenced by:  fneu  5578  elpreima  5879  fnopfv  5894  fnfvelrn  5896  funressnfv  28006  fnafvelrn  28047  afvco2  28054
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-11 1763  ax-ext 2423
This theorem depends on definitions:  df-bi 179  df-an 362  df-ex 1552  df-cleq 2435  df-clel 2438  df-fn 5486
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