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Theorem resundir 4833
 Description: Distributive law for restriction over union. (Contributed by NM, 23-Sep-2004.)
Assertion
Ref Expression
resundir ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))

Proof of Theorem resundir
StepHypRef Expression
1 indir 3325 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2 df-res 4551 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
3 df-res 4551 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
4 df-res 4551 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
53, 4uneq12i 3228 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
61, 2, 53eqtr4i 2170 1 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   = wceq 1331  Vcvv 2686   ∪ cun 3069   ∩ cin 3070   × cxp 4537   ↾ cres 4541 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-res 4551 This theorem is referenced by:  imaundir  4952  fvunsng  5614  fvsnun1  5617  fvsnun2  5618  fsnunfv  5621  fsnunres  5622  fseq1p1m1  9874  setsresg  11997  setscom  11999  setsslid  12009
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