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Theorem resindir 4717
Description: Class restriction distributes over intersection. (Contributed by NM, 18-Dec-2008.)
Assertion
Ref Expression
resindir ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))

Proof of Theorem resindir
StepHypRef Expression
1 inindir 3216 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∩ (𝐵 ∩ (𝐶 × V)))
2 df-res 4440 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
3 df-res 4440 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
4 df-res 4440 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
53, 4ineq12i 3197 . 2 ((𝐴𝐶) ∩ (𝐵𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∩ (𝐵 ∩ (𝐶 × V)))
61, 2, 53eqtr4i 2118 1 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∩ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1289  Vcvv 2619  cin 2996   × cxp 4426  cres 4430
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-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-in 3003  df-res 4440
This theorem is referenced by:  inimass  4835  fnreseql  5393
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