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Theorem reseq2 5089
Description: Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
Assertion
Ref Expression
reseq2  |-  ( A  =  B  ->  ( C  |`  A )  =  ( C  |`  B ) )

Proof of Theorem reseq2
StepHypRef Expression
1 xpeq1 4837 . . 3  |-  ( A  =  B  ->  ( A  X.  _V )  =  ( B  X.  _V ) )
21ineq2d 3641 . 2  |-  ( A  =  B  ->  ( C  i^i  ( A  X.  _V ) )  =  ( C  i^i  ( B  X.  _V ) ) )
3 df-res 4835 . 2  |-  ( C  |`  A )  =  ( C  i^i  ( A  X.  _V ) )
4 df-res 4835 . 2  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
52, 3, 43eqtr4g 2468 1  |-  ( A  =  B  ->  ( C  |`  A )  =  ( C  |`  B ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1405   _Vcvv 3059    i^i cin 3413    X. cxp 4821    |` cres 4825
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-v 3061  df-in 3421  df-opab 4454  df-xp 4829  df-res 4835
This theorem is referenced by:  reseq2i  5091  reseq2d  5094  resabs1  5122  resima2  5127  imaeq2  5153  resdisj  5254  relcoi1OLD  5353  fressnfv  6065  tfrlem1  7079  tfrlem9  7088  tfrlem11  7091  tfrlem12  7092  tfr2b  7099  tz7.44-1  7109  tz7.44-2  7110  tz7.44-3  7111  rdglem1  7118  fnfi  7832  fseqenlem1  8437  rtrclreclem4  13043  psgnprfval1  16871  gsumzaddlem  17258  gsumzaddlemOLD  17260  gsum2dlem2  17319  gsum2dOLD  17321  znunithash  18901  islinds  19136  lmbr2  20053  lmff  20095  kgencn2  20350  ptcmpfi  20606  tsmsgsum  20929  tsmsgsumOLD  20932  tsmsresOLD  20937  tsmsres  20938  tsmsf1o  20939  tsmsxplem1  20947  tsmsxp  20949  ustval  20997  xrge0gsumle  21630  xrge0tsms  21631  lmmbr2  21990  lmcau  22043  limcun  22591  jensen  23644  wilthlem2  23724  wilthlem3  23725  hhssnvt  26595  hhsssh  26599  foresf1o  27818  gsumle  28221  xrge0tsmsd  28228  esumsnf  28511  subfacp1lem3  29479  subfacp1lem5  29481  erdszelem1  29488  erdsze  29499  erdsze2lem2  29501  cvmscbv  29555  cvmshmeo  29568  cvmsss2  29571  ghomgrp  29882  dfpo2  29968  eldm3  29975  dfrdg2  30015  nofulllem4  30165  nofulllem5  30166  mbfresfi  31433  mzpcompact2lem  35045  seff  36037  fouriersw  37382
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