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Theorem reseq1 4549
 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 3125 . 2 (A = B → (A ∩ (𝐶 × V)) = (B ∩ (𝐶 × V)))
2 df-res 4300 . 2 (A𝐶) = (A ∩ (𝐶 × V))
3 df-res 4300 . 2 (B𝐶) = (B ∩ (𝐶 × V))
41, 2, 33eqtr4g 2094 1 (A = B → (A𝐶) = (B𝐶))
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1242  Vcvv 2551   ∩ cin 2910   × cxp 4286   ↾ cres 4290 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-res 4300 This theorem is referenced by:  reseq1i  4551  reseq1d  4554  imaeq1  4606  relcoi1  4792  tfr0  5878  tfrlemiex  5886
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