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Theorem funfni 5687
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 5684 . . 3  |-  ( F  Fn  A  ->  Fun  F )
21adantr 465 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  Fun  F )
3 fndm 5686 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
43eleq2d 2537 . . 3  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
54biimpar 485 . 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 661 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ph )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    e. wcel 1767   dom cdm 5005   Fun wfun 5588    Fn wfn 5589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-an 371  df-ex 1597  df-cleq 2459  df-clel 2462  df-fn 5597
This theorem is referenced by:  fneu  5691  elpreima  6008  fnopfv  6024  fnfvelrn  6029  funressnfv  32003  fnafvelrn  32044  afvco2  32051
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