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Theorem xpundi 4422
Description: Distributive law for cross 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 4377 . 2 (𝐴 × (𝐵𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐵𝐶))}
2 df-xp 4377 . . . 4 (𝐴 × 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)}
3 df-xp 4377 . . . 4 (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)}
42, 3uneq12i 3125 . . 3 ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)})
5 elun 3114 . . . . . . 7 (𝑦 ∈ (𝐵𝐶) ↔ (𝑦𝐵𝑦𝐶))
65anbi2i 445 . . . . . 6 ((𝑥𝐴𝑦 ∈ (𝐵𝐶)) ↔ (𝑥𝐴 ∧ (𝑦𝐵𝑦𝐶)))
7 andi 765 . . . . . 6 ((𝑥𝐴 ∧ (𝑦𝐵𝑦𝐶)) ↔ ((𝑥𝐴𝑦𝐵) ∨ (𝑥𝐴𝑦𝐶)))
86, 7bitri 182 . . . . 5 ((𝑥𝐴𝑦 ∈ (𝐵𝐶)) ↔ ((𝑥𝐴𝑦𝐵) ∨ (𝑥𝐴𝑦𝐶)))
98opabbii 3853 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐵𝐶))} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∨ (𝑥𝐴𝑦𝐶))}
10 unopab 3865 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐵) ∨ (𝑥𝐴𝑦𝐶))}
119, 10eqtr4i 2105 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐵𝐶))} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐵)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)})
124, 11eqtr4i 2105 . 2 ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 ∈ (𝐵𝐶))}
131, 12eqtr4i 2105 1 (𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∪ (𝐴 × 𝐶))
Colors of variables: wff set class
Syntax hints:  wa 102  wo 662   = wceq 1285  wcel 1434  cun 2972  {copab 3846   × cxp 4369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-v 2604  df-un 2978  df-opab 3848  df-xp 4377
This theorem is referenced by:  xpun  4427
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