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Theorem reseq1 4663
Description: Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
Assertion
Ref Expression
reseq1  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )

Proof of Theorem reseq1
StepHypRef Expression
1 ineq1 3178 . 2  |-  ( A  =  B  ->  ( A  i^i  ( C  X.  _V ) )  =  ( B  i^i  ( C  X.  _V ) ) )
2 df-res 4411 . 2  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
3 df-res 4411 . 2  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
41, 2, 33eqtr4g 2140 1  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1285   _Vcvv 2612    i^i cin 2983    X. cxp 4397    |` cres 4401
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 2614  df-in 2990  df-res 4411
This theorem is referenced by:  reseq1i  4665  reseq1d  4668  imaeq1  4722  relcoi1  4914  tfr0dm  6017  tfrlemiex  6026  tfr1onlemex  6042  tfr1onlemaccex  6043  tfrcllemsucaccv  6049  tfrcllembxssdm  6051  tfrcllemex  6055  tfrcllemaccex  6056  tfrcllemres  6057  pmresg  6361
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