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Theorem xpundi 5618
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 5559 . 2 (𝐴 × (𝐵𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐵𝐶))}
2 df-xp 5559 . . . 4 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
3 df-xp 5559 . . . 4 (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)}
42, 3uneq12i 4140 . . 3 ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)})
5 elun 4128 . . . . . . 7 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
65anbi2i 622 . . . . . 6 ((𝑥𝐴𝑦 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑦𝐶)))
7 andi 1003 . . . . . 6 ((𝑥𝐴 ∧ (𝑦𝐵𝑦𝐶)) ↔ ((𝑥𝐴𝑦𝐵) ∨ (𝑥𝐴𝑦𝐶)))
86, 7bitri 276 . . . . 5 ((𝑥𝐴𝑦 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑦𝐵) ∨ (𝑥𝐴𝑦𝐶)))
98opabbii 5129 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐵𝐶))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∨ (𝑥𝐴𝑦𝐶))}
10 unopab 5141 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∨ (𝑥𝐴𝑦𝐶))}
119, 10eqtr4i 2851 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐵𝐶))} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)})
124, 11eqtr4i 2851 . 2 ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐵𝐶))}
131, 12eqtr4i 2851 1 (𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 396  wo 843   = wceq 1530  wcel 2106  cun 3937  {copab 5124   × cxp 5551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-v 3501  df-un 3944  df-opab 5125  df-xp 5559
This theorem is referenced by:  xpun  5623  djuassen  9596  xpdjuen  9597  ustund  22745  bj-2upln1upl  34221
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