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Theorem fneq1 5218
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 5150 . . 3  |-  ( F  =  G  ->  ( Fun  F  <->  Fun  G ) )
2 dmeq 4746 . . . 4  |-  ( F  =  G  ->  dom  F  =  dom  G )
32eqeq1d 2149 . . 3  |-  ( F  =  G  ->  ( dom  F  =  A  <->  dom  G  =  A ) )
41, 3anbi12d 465 . 2  |-  ( F  =  G  ->  (
( Fun  F  /\  dom  F  =  A )  <-> 
( Fun  G  /\  dom  G  =  A ) ) )
5 df-fn 5133 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
6 df-fn 5133 . 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 1332   dom cdm 4546   Fun wfun 5124    Fn wfn 5125
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-fun 5132  df-fn 5133
This theorem is referenced by:  fneq1d  5220  fneq1i  5224  fn0  5249  feq1  5262  foeq1  5348  f1ocnv  5387  mpteqb  5518  eufnfv  5655  tfr0dm  6226  tfrlemiex  6235  tfr1onlemsucfn  6244  tfr1onlemsucaccv  6245  tfr1onlembxssdm  6247  tfr1onlembfn  6248  tfr1onlemex  6251  tfr1onlemaccex  6252  tfr1onlemres  6253  mapval2  6579  elixp2  6603  ixpfn  6605  elixpsn  6636  cc2lem  7097  cc3  7099
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