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Theorem funfni 5193
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 5190 . . 3  |-  ( F  Fn  A  ->  Fun  F )
21adantr 274 . 2  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  Fun  F )
3 fndm 5192 . . . 4  |-  ( F  Fn  A  ->  dom  F  =  A )
43eleq2d 2187 . . 3  |-  ( F  Fn  A  ->  ( B  e.  dom  F  <->  B  e.  A ) )
54biimpar 295 . 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 408 1  |-  ( ( F  Fn  A  /\  B  e.  A )  ->  ph )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    e. wcel 1465   dom cdm 4509   Fun wfun 5087    Fn wfn 5088
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1408  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-4 1472  ax-17 1491  ax-ial 1499  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-cleq 2110  df-clel 2113  df-fn 5096
This theorem is referenced by:  fneu  5197  fnbrfvb  5430  fvelrnb  5437  fvelimab  5445  fniinfv  5447  fvco2  5458  eqfnfv  5486  fndmdif  5493  fndmin  5495  elpreima  5507  fniniseg  5508  fniniseg2  5510  fnniniseg2  5511  rexsupp  5512  fnopfv  5518  fnfvelrn  5520  rexrn  5525  ralrn  5526  fsn2  5562  fnressn  5574  eufnfv  5616  rexima  5624  ralima  5625  fniunfv  5631  dff13  5637  foeqcnvco  5659  f1eqcocnv  5660  isocnv2  5681  isoini  5687  f1oiso  5695  fnovex  5772  suppssof1  5967  offveqb  5969  1stexg  6033  2ndexg  6034  smoiso  6167  rdgruledefgg  6240  rdgivallem  6246  frectfr  6265  frecrdg  6273  en1  6661  fnfi  6793  ordiso2  6888  slotex  11913
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