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Theorem fneq12 5291
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 5290 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 1348    Fn wfn 5193
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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-fun 5200  df-fn 5201
This theorem is referenced by:  tfrlem3ag  6288  tfrlem3a  6289  tfr1onlem3ag  6316  frecfnom  6380
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