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Theorem funfni 5423
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 5420 . . 3  |-  ( F  Fn  A  ->  Fun  F )
21adantr 451 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  Fun  F )
3 fndm 5422 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
43eleq2d 2425 . . 3  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
54biimpar 471 . 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 642 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1710   dom cdm 4768   Fun wfun 5328    Fn wfn 5329
This theorem is referenced by:  fneu  5427  elpreima  5725  fnopfv  5740  fnfvelrn  5742  funressnfv  27316  fnafvelrn  27357  afvco2  27364
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-11 1746  ax-ext 2339
This theorem depends on definitions:  df-bi 177  df-an 360  df-ex 1542  df-cleq 2351  df-clel 2354  df-fn 5337
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