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Theorem funfni 5508
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 5505 . . 3  |-  ( F  Fn  A  ->  Fun  F )
21adantr 452 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  Fun  F )
3 fndm 5507 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
43eleq2d 2475 . . 3  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
54biimpar 472 . 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 643 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1721   dom cdm 4841   Fun wfun 5411    Fn wfn 5412
This theorem is referenced by:  fneu  5512  elpreima  5813  fnopfv  5828  fnfvelrn  5830  funressnfv  27863  fnafvelrn  27904  afvco2  27911
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-11 1757  ax-ext 2389
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1548  df-cleq 2401  df-clel 2404  df-fn 5420
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