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Theorem reseq2 4814
Description: Equality theorem for restrictions. (Contributed by NM, 8-Aug-1994.)
Assertion
Ref Expression
reseq2 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))

Proof of Theorem reseq2
StepHypRef Expression
1 xpeq1 4553 . . 3 (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
21ineq2d 3277 . 2 (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3 df-res 4551 . 2 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
4 df-res 4551 . 2 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
52, 3, 43eqtr4g 2197 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  Vcvv 2686  cin 3070   × cxp 4537  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-opab 3990  df-xp 4545  df-res 4551
This theorem is referenced by:  reseq2i  4816  reseq2d  4819  resabs1  4848  resima2  4853  imaeq2  4877  resdisj  4967  relcoi1  5070  fressnfv  5607  tfrlem1  6205  tfrlem9  6216  tfr0dm  6219  tfrlemisucaccv  6222  tfrlemiubacc  6227  tfr1onlemsucaccv  6238  tfr1onlemubacc  6243  tfr1onlemaccex  6245  tfrcllemsucaccv  6251  tfrcllembxssdm  6253  tfrcllemubacc  6256  tfrcllemaccex  6258  tfrcllemres  6259  tfrcldm  6260  fnfi  6825  lmbr2  12383  lmff  12418
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