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Theorem reseq2 4817
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 4556 . . 3 (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
21ineq2d 3277 . 2 (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3 df-res 4554 . 2 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
4 df-res 4554 . 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 4540  cres 4544
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 3993  df-xp 4548  df-res 4554
This theorem is referenced by:  reseq2i  4819  reseq2d  4822  resabs1  4851  resima2  4856  imaeq2  4880  resdisj  4970  relcoi1  5073  fressnfv  5610  tfrlem1  6208  tfrlem9  6219  tfr0dm  6222  tfrlemisucaccv  6225  tfrlemiubacc  6230  tfr1onlemsucaccv  6241  tfr1onlemubacc  6246  tfr1onlemaccex  6248  tfrcllemsucaccv  6254  tfrcllembxssdm  6256  tfrcllemubacc  6259  tfrcllemaccex  6261  tfrcllemres  6262  tfrcldm  6263  fnfi  6828  lmbr2  12409  lmff  12444
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