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Theorem fneq12 5122
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 5121 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 1290    Fn wfn 5025
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071
This theorem depends on definitions:  df-bi 116  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-v 2624  df-un 3006  df-in 3008  df-ss 3015  df-sn 3458  df-pr 3459  df-op 3461  df-br 3854  df-opab 3908  df-rel 4461  df-cnv 4462  df-co 4463  df-dm 4464  df-fun 5032  df-fn 5033
This theorem is referenced by:  tfrlem3ag  6090  tfrlem3a  6091  tfr1onlem3ag  6118  frecfnom  6182
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