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Theorem resundir 4973
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 3422 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2 df-res 4687 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
3 df-res 4687 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
4 df-res 4687 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
53, 4uneq12i 3325 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
61, 2, 53eqtr4i 2236 1 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
Colors of variables: wff set class
Syntax hints:   = wceq 1373  Vcvv 2772  cun 3164  cin 3165   × cxp 4673  cres 4677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187
This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-un 3170  df-in 3172  df-res 4687
This theorem is referenced by:  imaundir  5096  fvunsng  5778  fvsnun1  5781  fvsnun2  5782  fsnunfv  5785  fsnunres  5786  fseq1p1m1  10216  setsresg  12870  setscom  12872  setsslid  12883
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