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Theorem xpundi 5094
Description: Distributive law for Cartesian product over union. Theorem 103 of [Suppes] p. 52. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
xpundi (𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶))

Proof of Theorem xpundi
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 5044 . 2 (𝐴 × (𝐵𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐵𝐶))}
2 df-xp 5044 . . . 4 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
3 df-xp 5044 . . . 4 (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)}
42, 3uneq12i 3727 . . 3 ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)})
5 elun 3715 . . . . . . 7 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
65anbi2i 726 . . . . . 6 ((𝑥𝐴𝑦 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑦𝐶)))
7 andi 907 . . . . . 6 ((𝑥𝐴 ∧ (𝑦𝐵𝑦𝐶)) ↔ ((𝑥𝐴𝑦𝐵) ∨ (𝑥𝐴𝑦𝐶)))
86, 7bitri 263 . . . . 5 ((𝑥𝐴𝑦 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑦𝐵) ∨ (𝑥𝐴𝑦𝐶)))
98opabbii 4649 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐵𝐶))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∨ (𝑥𝐴𝑦𝐶))}
10 unopab 4660 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∨ (𝑥𝐴𝑦𝐶))}
119, 10eqtr4i 2635 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐵𝐶))} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)})
124, 11eqtr4i 2635 . 2 ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐵𝐶))}
131, 12eqtr4i 2635 1 (𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wo 382  wa 383   = wceq 1475  wcel 1977  cun 3538  {copab 4642   × cxp 5036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545  df-opab 4644  df-xp 5044
This theorem is referenced by:  xpun  5099  xp2cda  8885  xpcdaen  8888  alephadd  9278  ustund  21835  bj-2upln1upl  32205
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