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Theorem xpundir 5747
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 5684 . 2 ((𝐴𝐵) × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)}
2 df-xp 5684 . . . 4 (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)}
3 df-xp 5684 . . . 4 (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}
42, 3uneq12i 4158 . . 3 ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
5 elun 4145 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65anbi1i 622 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑦𝐶))
7 andir 1006 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝑦𝐶) ↔ ((𝑥𝐴𝑦𝐶) ∨ (𝑥𝐵𝑦𝐶)))
86, 7bitri 274 . . . . 5 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶) ↔ ((𝑥𝐴𝑦𝐶) ∨ (𝑥𝐵𝑦𝐶)))
98opabbii 5216 . . . 4 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐶) ∨ (𝑥𝐵𝑦𝐶))}
10 unopab 5231 . . . 4 ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}) = {⟨𝑥, 𝑦⟩ ∣ ((𝑥𝐴𝑦𝐶) ∨ (𝑥𝐵𝑦𝐶))}
119, 10eqtr4i 2756 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
124, 11eqtr4i 2756 . 2 ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)}
131, 12eqtr4i 2756 1 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 394  wo 845   = wceq 1533  wcel 2098  cun 3942  {copab 5211   × cxp 5676
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-un 3949  df-opab 5212  df-xp 5684
This theorem is referenced by:  xpun  5751  resundi  5999  xpprsng  7149  naddasslem1  8715  xpfiOLD  9347  xp2dju  10206  alephadd  10607  hashxplem  14433  ustund  24175  cnmpopc  24898  poimirlem3  37229  poimirlem4  37230  poimirlem6  37232  poimirlem7  37233  poimirlem16  37242  poimirlem19  37245  fsuppssind  41963  pwssplit4  42657
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