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Theorem reseq2 5100
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 4851 . . 3  |-  ( A  =  B  ->  ( A  X.  _V )  =  ( B  X.  _V ) )
21ineq2d 3502 . 2  |-  ( A  =  B  ->  ( C  i^i  ( A  X.  _V ) )  =  ( C  i^i  ( B  X.  _V ) ) )
3 df-res 4849 . 2  |-  ( C  |`  A )  =  ( C  i^i  ( A  X.  _V ) )
4 df-res 4849 . 2  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
52, 3, 43eqtr4g 2461 1  |-  ( A  =  B  ->  ( C  |`  A )  =  ( C  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649   _Vcvv 2916    i^i cin 3279    X. cxp 4835    |` cres 4839
This theorem is referenced by:  reseq2i  5102  reseq2d  5105  resabs1  5134  resima2  5138  imaeq2  5158  resdisj  5257  relcoi1  5357  fressnfv  5879  tfrlem9  6605  tfrlem11  6608  tfrlem12  6609  tfr2b  6616  tz7.44-1  6623  tz7.44-2  6624  tz7.44-3  6625  rdglem1  6632  fnfi  7343  fseqenlem1  7861  gsumzaddlem  15481  gsum2d  15501  znunithash  16800  lmbr2  17277  lmff  17319  kgencn2  17542  ptcmpfi  17798  tsmsgsum  18121  tsmsres  18126  tsmsf1o  18127  tsmsxplem1  18135  tsmsxp  18137  ustval  18185  xrge0gsumle  18817  xrge0tsms  18818  lmmbr2  19165  lmcau  19218  limcun  19735  jensen  20780  wilthlem2  20805  wilthlem3  20806  hhssnvt  22718  hhsssh  22722  xrge0tsmsd  24176  esumsn  24409  subfacp1lem3  24821  subfacp1lem5  24823  erdszelem1  24830  erdsze  24841  erdsze2lem2  24843  cvmscbv  24898  cvmshmeo  24911  cvmsss2  24914  ghomgrp  25054  relexpcnv  25086  rtrclreclem.min  25100  dfpo2  25326  eldm3  25333  dfrdg2  25366  nofulllem4  25573  nofulllem5  25574  mbfresfi  26152  mzpcompact2lem  26698  islinds  27147  seff  27406
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-an 361  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-v 2918  df-in 3287  df-opab 4227  df-xp 4843  df-res 4849
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