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Theorem reseq1 4675
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 3183 . 2  |-  ( A  =  B  ->  ( A  i^i  ( C  X.  _V ) )  =  ( B  i^i  ( C  X.  _V ) ) )
2 df-res 4423 . 2  |-  ( A  |`  C )  =  ( A  i^i  ( C  X.  _V ) )
3 df-res 4423 . 2  |-  ( B  |`  C )  =  ( B  i^i  ( C  X.  _V ) )
41, 2, 33eqtr4g 2142 1  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1287   _Vcvv 2615    i^i cin 2987    X. cxp 4409    |` cres 4413
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 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-in 2994  df-res 4423
This theorem is referenced by:  reseq1i  4677  reseq1d  4680  imaeq1  4736  relcoi1  4928  tfr0dm  6041  tfrlemiex  6050  tfr1onlemex  6066  tfr1onlemaccex  6067  tfrcllemsucaccv  6073  tfrcllembxssdm  6075  tfrcllemex  6079  tfrcllemaccex  6080  tfrcllemres  6081  pmresg  6385
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