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Theorem fneq1d 5495
Description: Equality deduction for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypothesis
Ref Expression
fneq1d.1  |-  ( ph  ->  F  =  G )
Assertion
Ref Expression
fneq1d  |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  A ) )

Proof of Theorem fneq1d
StepHypRef Expression
1 fneq1d.1 . 2  |-  ( ph  ->  F  =  G )
2 fneq1 5493 . 2  |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
31, 2syl 16 1  |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    = wceq 1649    Fn wfn 5408
This theorem is referenced by:  fneq12d  5497  f1o00  5669  f1oprswap  5676  f1ompt  5850  fmpt2d  5857  f1ocnvd  6252  offn  6275  offval2  6281  ofrfval2  6282  caofinvl  6290  omxpenlem  7168  itunifn  8253  konigthlem  8399  seqof  11335  swrdlen  11725  fsumrev  12517  fsumshft  12518  prdsdsfn  13642  imasdsfn  13695  xpscfn  13739  cidfn  13859  comffn  13886  isoval  13945  invf1o  13949  brssc  13969  cofucl  14040  1stfcl  14249  2ndfcl  14250  prfcl  14255  evlfcl  14274  curf1cl  14280  curfcl  14284  hofcl  14311  yonedainv  14333  grpinvf1o  14816  srngf1o  15897  neif  17119  fmf  17930  fncpn  19772  grpoinvf  21781  kbass2  23573  f1o3d  23994  offval2f  24033  pstmxmet  24245  ofcfn  24436  ofcfval2  24440  fprodshft  25253  fprodrev  25254  cnambfre  26154  sdclem2  26336  hbtlem7  27197  pmtrrn  27267  pmtrfrn  27268  addrfn  27544  subrfn  27545  mulvfn  27546  bnj941  28849  diafn  31517  dibfna  31637  dicfnN  31666  dihf11lem  31749  mapd1o  32131  hdmapfnN  32315  hgmapfnN  32374
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-br 4173  df-opab 4227  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-fun 5415  df-fn 5416
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