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Theorem reseq1i 4815
Description: Equality inference for restrictions. (Contributed by NM, 21-Oct-2014.)
Hypothesis
Ref Expression
reseqi.1  |-  A  =  B
Assertion
Ref Expression
reseq1i  |-  ( A  |`  C )  =  ( B  |`  C )

Proof of Theorem reseq1i
StepHypRef Expression
1 reseqi.1 . 2  |-  A  =  B
2 reseq1 4813 . 2  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
31, 2ax-mp 5 1  |-  ( A  |`  C )  =  ( B  |`  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1331    |` cres 4541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-in 3077  df-res 4551
This theorem is referenced by:  reseq12i  4817  resindm  4861  resmpt  4867  resmpt3  4868  resmptf  4869  opabresid  4872  rescnvcnv  5001  coires1  5056  fcoi1  5303  fvsnun1  5617  fvsnun2  5618  resoprab  5867  resmpo  5869  ofmres  6034  f1stres  6057  f2ndres  6058  df1st2  6116  df2nd2  6117  dftpos2  6158  tfr2a  6218  freccllem  6299  frecfcllem  6301  frecsuclem  6303  djuf1olemr  6939  divfnzn  9425  cnmptid  12464  xmsxmet2  12646  msmet2  12647
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