ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fneq1d Unicode version

Theorem fneq1d 5104
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 5102 . 2  |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
31, 2syl 14 1  |-  ( ph  ->  ( F  Fn  A  <->  G  Fn  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 103    = wceq 1289    Fn wfn 5010
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-opab 3900  df-rel 4445  df-cnv 4446  df-co 4447  df-dm 4448  df-fun 5017  df-fn 5018
This theorem is referenced by:  fneq12d  5106  f1o00  5288  f1ompt  5450  fmpt2d  5460  f1ocnvd  5846  offval2  5870  ofrfval2  5871  caofinvl  5877  f1od2  6000
  Copyright terms: Public domain W3C validator