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Theorem fneq12 5186
Description: Equality theorem for function predicate with domain. (Contributed by Thierry Arnoux, 31-Jan-2017.)
Assertion
Ref Expression
fneq12  |-  ( ( F  =  G  /\  A  =  B )  ->  ( F  Fn  A  <->  G  Fn  B ) )

Proof of Theorem fneq12
StepHypRef Expression
1 simpl 108 . 2  |-  ( ( F  =  G  /\  A  =  B )  ->  F  =  G )
2 simpr 109 . 2  |-  ( ( F  =  G  /\  A  =  B )  ->  A  =  B )
31, 2fneq12d 5185 1  |-  ( ( F  =  G  /\  A  =  B )  ->  ( F  Fn  A  <->  G  Fn  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    = wceq 1316    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:  tfrlem3ag  6174  tfrlem3a  6175  tfr1onlem3ag  6202  frecfnom  6266
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