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Theorem xpundir 5746
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 5683 . 2 ((𝐴𝐵) × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)}
2 df-xp 5683 . . . 4 (𝐴 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)}
3 df-xp 5683 . . . 4 (𝐵 × 𝐶) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)}
42, 3uneq12i 4162 . . 3 ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
5 elun 4149 . . . . . . 7 (𝑥 ∈ (𝐴𝐵) ↔ (𝑥𝐴𝑥𝐵))
65anbi1i 625 . . . . . 6 ((𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶) ↔ ((𝑥𝐴𝑥𝐵) ∧ 𝑦𝐶))
7 andir 1008 . . . . . 6 (((𝑥𝐴𝑥𝐵) ∧ 𝑦𝐶) ↔ ((𝑥𝐴𝑦𝐶) ∨ (𝑥𝐵𝑦𝐶)))
86, 7bitri 275 . . . . 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 2764 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)} = ({⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝐶)} ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐵𝑦𝐶)})
124, 11eqtr4i 2764 . 2 ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ (𝐴𝐵) ∧ 𝑦𝐶)}
131, 12eqtr4i 2764 1 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wa 397  wo 846   = wceq 1542  wcel 2107  cun 3947  {copab 5211   × cxp 5675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-opab 5212  df-xp 5683
This theorem is referenced by:  xpun  5750  resundi  5996  xpprsng  7138  naddasslem1  8693  xpfiOLD  9318  xp2dju  10171  alephadd  10572  hashxplem  14393  ustund  23726  cnmpopc  24444  poimirlem3  36491  poimirlem4  36492  poimirlem6  36494  poimirlem7  36495  poimirlem16  36504  poimirlem19  36507  fsuppssind  41165  pwssplit4  41831
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