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Theorem resundi 4652
 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 4424 . . . 4 ((𝐵𝐶) × V) = ((𝐵 × V) ∪ (𝐶 × V))
21ineq2i 3162 . . 3 (𝐴 ∩ ((𝐵𝐶) × V)) = (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V)))
3 indi 3211 . . 3 (𝐴 ∩ ((𝐵 × V) ∪ (𝐶 × V))) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
42, 3eqtri 2076 . 2 (𝐴 ∩ ((𝐵𝐶) × V)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
5 df-res 4384 . 2 (𝐴 ↾ (𝐵𝐶)) = (𝐴 ∩ ((𝐵𝐶) × V))
6 df-res 4384 . . 3 (𝐴𝐵) = (𝐴 ∩ (𝐵 × V))
7 df-res 4384 . . 3 (𝐴𝐶) = (𝐴 ∩ (𝐶 × V))
86, 7uneq12i 3122 . 2 ((𝐴𝐵) ∪ (𝐴𝐶)) = ((𝐴 ∩ (𝐵 × V)) ∪ (𝐴 ∩ (𝐶 × V)))
94, 5, 83eqtr4i 2086 1 (𝐴 ↾ (𝐵𝐶)) = ((𝐴𝐵) ∪ (𝐴𝐶))
 Colors of variables: wff set class Syntax hints:   = wceq 1259  Vcvv 2574   ∪ cun 2942   ∩ cin 2943   × cxp 4370   ↾ cres 4374 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038 This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-un 2949  df-in 2951  df-opab 3846  df-xp 4378  df-res 4384 This theorem is referenced by:  imaundi  4763  relresfld  4874  relcoi1  4876  resasplitss  5096  fseq1p1m1  9057
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