ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  reseq2 GIF version

Theorem reseq2 4696
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 4442 . . 3 (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
21ineq2d 3199 . 2 (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3 df-res 4440 . 2 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
4 df-res 4440 . 2 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
52, 3, 43eqtr4g 2145 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1289  Vcvv 2619  cin 2996   × cxp 4426  cres 4430
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 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-opab 3892  df-xp 4434  df-res 4440
This theorem is referenced by:  reseq2i  4698  reseq2d  4701  resabs1  4729  resima2  4733  imaeq2  4757  resdisj  4846  relcoi1  4949  fressnfv  5468  tfrlem1  6055  tfrlem9  6066  tfr0dm  6069  tfrlemisucaccv  6072  tfrlemiubacc  6077  tfr1onlemsucaccv  6088  tfr1onlemubacc  6093  tfr1onlemaccex  6095  tfrcllemsucaccv  6101  tfrcllembxssdm  6103  tfrcllemubacc  6106  tfrcllemaccex  6108  tfrcllemres  6109  tfrcldm  6110  fnfi  6625
  Copyright terms: Public domain W3C validator