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Theorem resres 4567
Description: The restriction of a restriction. (Contributed by NM, 27-Mar-2008.)
Assertion
Ref Expression
resres ((AB) ↾ 𝐶) = (A ↾ (B𝐶))

Proof of Theorem resres
StepHypRef Expression
1 df-res 4300 . 2 ((AB) ↾ 𝐶) = ((AB) ∩ (𝐶 × V))
2 df-res 4300 . . 3 (AB) = (A ∩ (B × V))
32ineq1i 3128 . 2 ((AB) ∩ (𝐶 × V)) = ((A ∩ (B × V)) ∩ (𝐶 × V))
4 xpindir 4415 . . . 4 ((B𝐶) × V) = ((B × V) ∩ (𝐶 × V))
54ineq2i 3129 . . 3 (A ∩ ((B𝐶) × V)) = (A ∩ ((B × V) ∩ (𝐶 × V)))
6 df-res 4300 . . 3 (A ↾ (B𝐶)) = (A ∩ ((B𝐶) × V))
7 inass 3141 . . 3 ((A ∩ (B × V)) ∩ (𝐶 × V)) = (A ∩ ((B × V) ∩ (𝐶 × V)))
85, 6, 73eqtr4ri 2068 . 2 ((A ∩ (B × V)) ∩ (𝐶 × V)) = (A ↾ (B𝐶))
91, 3, 83eqtri 2061 1 ((AB) ↾ 𝐶) = (A ↾ (B𝐶))
Colors of variables: wff set class
Syntax hints:   = 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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-opab 3810  df-xp 4294  df-rel 4295  df-res 4300
This theorem is referenced by:  rescom  4579  resabs1  4583  resima2  4587  resmpt3  4600  resdisj  4694  rescnvcnv  4726  funimaexg  4926  fresin  5011  resdif  5091
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