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Theorem reseq2 4635
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 4385 . . 3 (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
21ineq2d 3174 . 2 (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3 df-res 4383 . 2 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
4 df-res 4383 . 2 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
52, 3, 43eqtr4g 2139 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1285  Vcvv 2602  cin 2973   × cxp 4369  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-opab 3848  df-xp 4377  df-res 4383
This theorem is referenced by:  reseq2i  4637  reseq2d  4640  resabs1  4668  resima2  4672  imaeq2  4694  resdisj  4781  relcoi1  4879  fressnfv  5382  tfrlem1  5957  tfrlem9  5968  tfr0dm  5971  tfrlemisucaccv  5974  tfrlemiubacc  5979  tfr1onlemsucaccv  5990  tfr1onlemubacc  5995  tfr1onlemaccex  5997  tfrcllemsucaccv  6003  tfrcllembxssdm  6005  tfrcllemubacc  6008  tfrcllemaccex  6010  tfrcllemres  6011  tfrcldm  6012  fnfi  6446
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