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Theorem resundi 4981
 Description: Distributive law for restriction over union. Theorem 31 of [Suppes] p. 65. (Contributed by set.mm contributors, 30-Sep-2002.)
Assertion
Ref Expression
resundi (A (BC)) = ((A B) ∪ (A C))

Proof of Theorem resundi
StepHypRef Expression
1 xpundir 4833 . . . 4 ((BC) × V) = ((B × V) ∪ (C × V))
21ineq2i 3454 . . 3 (A ∩ ((BC) × V)) = (A ∩ ((B × V) ∪ (C × V)))
3 indi 3501 . . 3 (A ∩ ((B × V) ∪ (C × V))) = ((A ∩ (B × V)) ∪ (A ∩ (C × V)))
42, 3eqtri 2373 . 2 (A ∩ ((BC) × V)) = ((A ∩ (B × V)) ∪ (A ∩ (C × V)))
5 df-res 4788 . 2 (A (BC)) = (A ∩ ((BC) × V))
6 df-res 4788 . . 3 (A B) = (A ∩ (B × V))
7 df-res 4788 . . 3 (A C) = (A ∩ (C × V))
86, 7uneq12i 3416 . 2 ((A B) ∪ (A C)) = ((A ∩ (B × V)) ∪ (A ∩ (C × V)))
94, 5, 83eqtr4i 2383 1 (A (BC)) = ((A B) ∪ (A C))
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642  Vcvv 2859   ∪ cun 3207   ∩ cin 3208   × cxp 4770   ↾ cres 4774 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-opab 4623  df-xp 4784  df-res 4788 This theorem is referenced by:  imaundi  5039
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