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Theorem resindi 5952
Description: Class restriction distributes over intersection. (Contributed by FL, 6-Oct-2008.)
Assertion
Ref Expression
resindi (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))

Proof of Theorem resindi
StepHypRef Expression
1 xpindir 5789 . . . 4 ((𝐵𝐶) × V) = ((𝐵 × V) ∩ (𝐶 × V))
21ineq2i 4168 . . 3 (𝐴 ∩ ((𝐵𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V)))
3 inindi 4185 . . 3 (𝐴 ∩ ((𝐵 × V) ∩ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
42, 3eqtri 2766 . 2 (𝐴 ∩ ((𝐵𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
5 df-res 5644 . 2 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ∩ ((𝐵𝐶) × V))
6 df-res 5644 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
7 df-res 5644 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
86, 7ineq12i 4169 . 2 ((𝐴𝐵) ∩ (𝐴𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∩ (𝐴 ∩ (𝐶 × V)))
94, 5, 83eqtr4i 2776 1 (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∩ (𝐴𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  Vcvv 3444  cin 3908   × cxp 5630  cres 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2709  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5167  df-xp 5638  df-rel 5639  df-res 5644
This theorem is referenced by:  resindm  5985  gsum2dlem2  19707  fnresin  31368  fressupp  31429  disjresdisj  36630
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