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Theorem resres 4725
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 4450 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
2 df-res 4450 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
32ineq1i 3197 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V))
4 xpindir 4572 . . . 4 ((𝐵𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
54ineq2i 3198 . . 3 (𝐴 ∩ ((𝐵𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
6 df-res 4450 . . 3 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ∩ ((𝐵𝐶) × V))
7 inass 3210 . . 3 ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
85, 6, 73eqtr4ri 2119 . 2 ((𝐴 ∩ (𝐵 × V)) ∩ (𝐶 × V)) = (𝐴 ↾ (𝐵𝐶))
91, 3, 83eqtri 2112 1 ((𝐴𝐵) ↾ 𝐶) = (𝐴 ↾ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1289  Vcvv 2619  cin 2998   × cxp 4436  cres 4440
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957  ax-pow 4009  ax-pr 4036
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-pw 3431  df-sn 3452  df-pr 3453  df-op 3455  df-opab 3900  df-xp 4444  df-rel 4445  df-res 4450
This theorem is referenced by:  rescom  4738  resabs1  4742  resima2  4746  resmpt3  4761  resdisj  4859  rescnvcnv  4893  funimaexg  5098  fresin  5189  resdif  5275  pmresg  6431  setsidn  11539
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