MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  fneq1i Unicode version

Theorem fneq1i 5338
Description: Equality inference for function predicate with domain. (Contributed by Paul Chapman, 22-Jun-2011.)
Hypothesis
Ref Expression
fneq1i.1  |-  F  =  G
Assertion
Ref Expression
fneq1i  |-  ( F  Fn  A  <->  G  Fn  A )

Proof of Theorem fneq1i
StepHypRef Expression
1 fneq1i.1 . 2  |-  F  =  G
2 fneq1 5333 . 2  |-  ( F  =  G  ->  ( F  Fn  A  <->  G  Fn  A ) )
31, 2ax-mp 8 1  |-  ( F  Fn  A  <->  G  Fn  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    = wceq 1623    Fn wfn 5250
This theorem is referenced by:  fnunsn  5351  fnopabg  5367  f1oun  5492  f1oi  5511  f1osn  5513  ovid  5964  curry1  6210  curry2  6213  tfrlem10  6403  tfr1  6413  seqomlem2  6463  seqomlem3  6464  seqomlem4  6465  fnseqom  6467  abianfp  6471  unblem4  7112  r1fnon  7439  alephfnon  7692  alephfplem4  7734  alephfp  7735  cfsmolem  7896  infpssrlem3  7931  compssiso  8000  hsmexlem5  8056  axdclem2  8147  wunex2  8360  wuncval2  8369  om2uzrani  11015  om2uzf1oi  11016  uzrdglem  11020  uzrdgfni  11021  uzrdg0i  11022  hashkf  11339  dmaf  13881  cdaf  13882  prdsinvlem  14603  pws1  15399  ovolunlem1  18856  0plef  19027  0pledm  19028  itg1ge0  19041  itg1addlem4  19054  mbfi1fseqlem5  19074  itg2addlem  19113  qaa  19703  ghgrp  21035  0vfval  21162  mptfnf  23226  xrge0pluscn  23322  eupap1  23900  wfrlem5  24260  wfrlem13  24268  frrlem5  24285  fullfunfnv  24484  trooo  25394  trinv  25395  neibastop2lem  26309  bnj927  28800  bnj535  28922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-fun 5257  df-fn 5258
  Copyright terms: Public domain W3C validator