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Theorem resundir 4982
 Description: Distributive law for restriction over union. (Contributed by set.mm contributors, 23-Sep-2004.)
Assertion
Ref Expression
resundir ((AB) C) = ((A C) ∪ (B C))

Proof of Theorem resundir
StepHypRef Expression
1 indir 3503 . 2 ((AB) ∩ (C × V)) = ((A ∩ (C × V)) ∪ (B ∩ (C × V)))
2 df-res 4788 . 2 ((AB) C) = ((AB) ∩ (C × V))
3 df-res 4788 . . 3 (A C) = (A ∩ (C × V))
4 df-res 4788 . . 3 (B C) = (B ∩ (C × V))
53, 4uneq12i 3416 . 2 ((A C) ∪ (B C)) = ((A ∩ (C × V)) ∪ (B ∩ (C × V)))
61, 2, 53eqtr4i 2383 1 ((AB) C) = ((A C) ∪ (B 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-mp 5  ax-1 6  ax-2 7  ax-3 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-res 4788 This theorem is referenced by:  imaundir  5040  fvunsn  5444  fvsnun1  5447  fvsnun2  5448
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