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Theorem reseq2 4550
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 4302 . . 3 (A = B → (A × V) = (B × V))
21ineq2d 3132 . 2 (A = B → (𝐶 ∩ (A × V)) = (𝐶 ∩ (B × V)))
3 df-res 4300 . 2 (𝐶A) = (𝐶 ∩ (A × V))
4 df-res 4300 . 2 (𝐶B) = (𝐶 ∩ (B × V))
52, 3, 43eqtr4g 2094 1 (A = B → (𝐶A) = (𝐶B))
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-opab 3810  df-xp 4294  df-res 4300
This theorem is referenced by:  reseq2i  4552  reseq2d  4555  resabs1  4583  resima2  4587  imaeq2  4607  resdisj  4694  relcoi1  4792  fressnfv  5293  tfrlem1  5864  tfrlem9  5876  tfr0  5878  tfrlemisucaccv  5880  tfrlemiubacc  5885
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