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Theorem fneq1 5625
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 5563 . . 3  |-  ( F  =  G  ->  ( Fun  F  <->  Fun  G ) )
2 dmeq 4997 . . . 4  |-  ( F  =  G  ->  dom  F  =  dom  G )
32eqeq1d 2430 . . 3  |-  ( F  =  G  ->  ( dom  F  =  A  <->  dom  G  =  A ) )
41, 3anbi12d 715 . 2  |-  ( F  =  G  ->  (
( Fun  F  /\  dom  F  =  A )  <-> 
( Fun  G  /\  dom  G  =  A ) ) )
5 df-fn 5547 . 2  |-  ( F  Fn  A  <->  ( Fun  F  /\  dom  F  =  A ) )
6 df-fn 5547 . 2  |-  ( G  Fn  A  <->  ( Fun  G  /\  dom  G  =  A ) )
74, 5, 63bitr4g 291 1  |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437   dom cdm 4796   Fun wfun 5538    Fn wfn 5539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-rab 2723  df-v 3024  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-op 3948  df-br 4367  df-opab 4426  df-rel 4803  df-cnv 4804  df-co 4805  df-dm 4806  df-fun 5546  df-fn 5547
This theorem is referenced by:  fneq1d  5627  fneq1i  5631  fn0  5656  feq1  5671  foeq1  5749  f1ocnv  5786  dffn5  5870  mpteqb  5924  fnsnb  6042  fnprb  6082  eufnfv  6098  wfrlem1  6990  wfrlem3a  6993  wfrlem15  7005  tfrlem12  7062  mapval2  7456  elixp2  7481  ixpfn  7483  elixpsn  7516  inf3lem6  8091  aceq3lem  8502  dfac4  8504  dfacacn  8522  axcc2lem  8817  axcc3  8819  seqof  12220  ccatvalfn  12674  cshword  12839  0csh0  12841  rrgsupp  18458  elpt  20529  elptr  20530  ptcmplem3  21011  prdsxmslem2  21486  bnj62  29478  bnj976  29541  bnj66  29623  bnj124  29634  bnj607  29679  bnj873  29687  bnj1234  29774  bnj1463  29816  frrlem1  30465  fnchoice  37266  dfafn5b  38476  cshword2  38791  rngchomffvalALTV  39588
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