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Theorem reseq1 4529
Description: Equality theorem for restrictions. (Contributed by NM, 7-Aug-1994.)
Assertion
Ref Expression
reseq1 (A = B → (A𝐶) = (B𝐶))

Proof of Theorem reseq1
StepHypRef Expression
1 ineq1 3104 . 2 (A = B → (A ∩ (𝐶 × V)) = (B ∩ (𝐶 × V)))
2 df-res 4280 . 2 (A𝐶) = (A ∩ (𝐶 × V))
3 df-res 4280 . 2 (B𝐶) = (B ∩ (𝐶 × V))
41, 2, 33eqtr4g 2075 1 (A = B → (A𝐶) = (B𝐶))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1226  Vcvv 2531  cin 2889   × cxp 4266  cres 4270
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000
This theorem depends on definitions:  df-bi 110  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-v 2533  df-in 2897  df-res 4280
This theorem is referenced by:  reseq1i  4531  reseq1d  4534  imaeq1  4586  relcoi1  4772  tfrlemiex  5862
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