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Theorem xpundir 5206
Description: Distributive law for Cartesian product over union. Similar to Theorem 103 of [Suppes] p. 52. (Contributed by NM, 30-Sep-2002.)
Assertion
Ref Expression
xpundir ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))

Proof of Theorem xpundir
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-xp 5149 . 2 ((𝐴𝐵) × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)}
2 df-xp 5149 . . . 4 (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)}
3 df-xp 5149 . . . 4 (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}
42, 3uneq12i 3798 . . 3 ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
5 elun 3786 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65anbi1i 731 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑦𝐶))
7 andir 930 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝑦𝐶) ↔ ((𝑥𝐴𝑦𝐶) ∨ (𝑥𝐵𝑦𝐶)))
86, 7bitri 264 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶) ↔ ((𝑥𝐴𝑦𝐶) ∨ (𝑥𝐵𝑦𝐶)))
98opabbii 4750 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐶) ∨ (𝑥𝐵𝑦𝐶))}
10 unopab 4761 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐶) ∨ (𝑥𝐵𝑦𝐶))}
119, 10eqtr4i 2676 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
124, 11eqtr4i 2676 . 2 ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)}
131, 12eqtr4i 2676 1 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wo 382  wa 383   = wceq 1523  wcel 2030  cun 3605  {copab 4745   × cxp 5141
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-v 3233  df-un 3612  df-opab 4746  df-xp 5149
This theorem is referenced by:  xpun  5210  resundi  5445  xpfi  8272  cdaassen  9042  hashxplem  13258  ustund  22072  cnmpt2pc  22774  poimirlem3  33542  poimirlem4  33543  poimirlem6  33545  poimirlem7  33546  poimirlem16  33555  poimirlem19  33558  pwssplit4  37976  xpprsng  42435
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