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Theorem resundi 4878
 Description: Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
resundi (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))

Proof of Theorem resundi
StepHypRef Expression
1 xpundir 4642 . . . 4 ((𝐵𝐶) × V) = ((𝐵 × V) ∪ (𝐶 × V))
21ineq2i 3305 . . 3 (𝐴 ∩ ((𝐵𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V)))
3 indi 3354 . . 3 (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
42, 3eqtri 2178 . 2 (𝐴 ∩ ((𝐵𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
5 df-res 4597 . 2 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ∩ ((𝐵𝐶) × V))
6 df-res 4597 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
7 df-res 4597 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
86, 7uneq12i 3259 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
94, 5, 83eqtr4i 2188 1 (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 Colors of variables: wff set class Syntax hints:   = wceq 1335  Vcvv 2712   ∪ cun 3100   ∩ cin 3101   × cxp 4583   ↾ cres 4587 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-un 3106  df-in 3108  df-opab 4026  df-xp 4591  df-res 4597 This theorem is referenced by:  imaundi  4997  relresfld  5114  relcoi1  5116  resasplitss  5348  fnsnsplitss  5665  fnsnsplitdc  6449  fnfi  6878  fseq1p1m1  9989  resunimafz0  10695
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