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Theorem reseq1i 4636
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 4634 . 2  |-  ( A  =  B  ->  ( A  |`  C )  =  ( B  |`  C ) )
31, 2ax-mp 7 1  |-  ( A  |`  C )  =  ( B  |`  C )
Colors of variables: wff set class
Syntax hints:    = wceq 1285    |` cres 4373
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 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-in 2980  df-res 4383
This theorem is referenced by:  reseq12i  4638  resindm  4680  resmpt  4686  resmpt3  4687  opabresid  4689  rescnvcnv  4813  coires1  4868  fcoi1  5101  fvsnun1  5392  fvsnun2  5393  resoprab  5628  resmpt2  5630  ofmres  5794  f1stres  5817  f2ndres  5818  df1st2  5871  df2nd2  5872  dftpos2  5910  tfr2a  5970  freccllem  6051  frecfcllem  6053  frecsuclem  6055  divfnzn  8787
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