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Theorem resundir 4626
 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 3186 . 2 ((𝐴𝐵) ∩ (𝐶 × V)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
2 df-res 4357 . 2 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐵) ∩ (𝐶 × V))
3 df-res 4357 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
4 df-res 4357 . . 3 (𝐵𝐶) = (𝐵 ∩ (𝐶 × V))
53, 4uneq12i 3095 . 2 ((𝐴𝐶) ∪ (𝐵𝐶)) = ((𝐴 ∩ (𝐶 × V)) ∪ (𝐵 ∩ (𝐶 × V)))
61, 2, 53eqtr4i 2070 1 ((𝐴𝐵) ↾ 𝐶) = ((𝐴𝐶) ∪ (𝐵𝐶))
 Colors of variables: wff set class Syntax hints:   = wceq 1243  Vcvv 2557   ∪ cun 2915   ∩ cin 2916   × cxp 4343   ↾ cres 4347 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-res 4357 This theorem is referenced by:  imaundir  4737  fvunsng  5357  fvsnun1  5360  fvsnun2  5361  fsnunfv  5363  fsnunres  5364  fseq1p1m1  8956
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