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Theorem reseq2 4655
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 4405 . . 3  |-  ( A  =  B  ->  ( A  X.  _V )  =  ( B  X.  _V ) )
21ineq2d 3183 . 2  |-  ( A  =  B  ->  ( C  i^i  ( A  X.  _V ) )  =  ( C  i^i  ( B  X.  _V ) ) )
3 df-res 4403 . 2  |-  ( C  |`  A )  =  ( C  i^i  ( A  X.  _V ) )
4 df-res 4403 . 2  |-  ( C  |`  B )  =  ( C  i^i  ( B  X.  _V ) )
52, 3, 43eqtr4g 2140 1  |-  ( A  =  B  ->  ( C  |`  A )  =  ( C  |`  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   _Vcvv 2610    i^i cin 2981    X. cxp 4389    |` cres 4393
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 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2612  df-in 2988  df-opab 3860  df-xp 4397  df-res 4403
This theorem is referenced by:  reseq2i  4657  reseq2d  4660  resabs1  4688  resima2  4692  imaeq2  4714  resdisj  4801  relcoi1  4899  fressnfv  5403  tfrlem1  5978  tfrlem9  5989  tfr0dm  5992  tfrlemisucaccv  5995  tfrlemiubacc  6000  tfr1onlemsucaccv  6011  tfr1onlemubacc  6016  tfr1onlemaccex  6018  tfrcllemsucaccv  6024  tfrcllembxssdm  6026  tfrcllemubacc  6029  tfrcllemaccex  6031  tfrcllemres  6032  tfrcldm  6033  fnfi  6479
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