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Theorem fneq1 5181
Description: Equality theorem for function predicate with domain. (Contributed by NM, 1-Aug-1994.)
Assertion
Ref Expression
fneq1  |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )

Proof of Theorem fneq1
StepHypRef Expression
1 funeq 5113 . . 3  |-  ( F  =  G  ->  ( Fun  F  <->  Fun  G ) )
2 dmeq 4709 . . . 4  |-  ( F  =  G  ->  dom  F  =  dom  G )
32eqeq1d 2126 . . 3  |-  ( F  =  G  ->  ( dom  F  =  A  <->  dom  G  =  A ) )
41, 3anbi12d 464 . 2  |-  ( F  =  G  ->  (
( Fun  F  /\  dom  F  =  A )  <-> 
( Fun  G  /\  dom  G  =  A ) ) )
5 df-fn 5096 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
6 df-fn 5096 . 2  |-  ( G  Fn  A  <->  ( Fun  G  /\  dom  G  =  A ) )
74, 5, 63bitr4g 222 1  |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316   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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-pr 3504  df-op 3506  df-br 3900  df-opab 3960  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-fun 5095  df-fn 5096
This theorem is referenced by:  fneq1d  5183  fneq1i  5187  fn0  5212  feq1  5225  foeq1  5311  f1ocnv  5348  mpteqb  5479  eufnfv  5616  tfr0dm  6187  tfrlemiex  6196  tfr1onlemsucfn  6205  tfr1onlemsucaccv  6206  tfr1onlembxssdm  6208  tfr1onlembfn  6209  tfr1onlemex  6212  tfr1onlemaccex  6213  tfr1onlemres  6214  mapval2  6540  elixp2  6564  ixpfn  6566  elixpsn  6597
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