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Theorem reseq2 4920
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 4658 . . 3 (𝐴 = 𝐵 → (𝐴 × V) = (𝐵 × V))
21ineq2d 3351 . 2 (𝐴 = 𝐵 → (𝐶 ∩ (𝐴 × V)) = (𝐶 ∩ (𝐵 × V)))
3 df-res 4656 . 2 (𝐶𝐴) = (𝐶 ∩ (𝐴 × V))
4 df-res 4656 . 2 (𝐶𝐵) = (𝐶 ∩ (𝐵 × V))
52, 3, 43eqtr4g 2247 1 (𝐴 = 𝐵 → (𝐶𝐴) = (𝐶𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1364  Vcvv 2752  cin 3143   × cxp 4642  cres 4646
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-in 3150  df-opab 4080  df-xp 4650  df-res 4656
This theorem is referenced by:  reseq2i  4922  reseq2d  4925  resabs1  4954  resima2  4959  imaeq2  4984  resdisj  5075  relcoi1  5178  fressnfv  5724  tfrlem1  6334  tfrlem9  6345  tfr0dm  6348  tfrlemisucaccv  6351  tfrlemiubacc  6356  tfr1onlemsucaccv  6367  tfr1onlemubacc  6372  tfr1onlemaccex  6374  tfrcllemsucaccv  6380  tfrcllembxssdm  6382  tfrcllemubacc  6385  tfrcllemaccex  6387  tfrcllemres  6388  tfrcldm  6389  fnfi  6967  lmbr2  14191  lmff  14226
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