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Theorem xpun 5137
Description: The Cartesian product of two unions. (Contributed by NM, 12-Aug-2004.)
Assertion
Ref Expression
xpun ((𝐴𝐵) × (𝐶𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))

Proof of Theorem xpun
StepHypRef Expression
1 xpundi 5132 . 2 ((𝐴𝐵) × (𝐶𝐷)) = (((𝐴𝐵) × 𝐶) ∪ ((𝐴𝐵) × 𝐷))
2 xpundir 5133 . . 3 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∪ (𝐵 × 𝐶))
3 xpundir 5133 . . 3 ((𝐴𝐵) × 𝐷) = ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))
42, 3uneq12i 3743 . 2 (((𝐴𝐵) × 𝐶) ∪ ((𝐴𝐵) × 𝐷)) = (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷)))
5 un4 3751 . 2 (((𝐴 × 𝐶) ∪ (𝐵 × 𝐶)) ∪ ((𝐴 × 𝐷) ∪ (𝐵 × 𝐷))) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
61, 4, 53eqtri 2647 1 ((𝐴𝐵) × (𝐶𝐷)) = (((𝐴 × 𝐶) ∪ (𝐴 × 𝐷)) ∪ ((𝐵 × 𝐶) ∪ (𝐵 × 𝐷)))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1480  cun 3553   × cxp 5072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-v 3188  df-un 3560  df-opab 4674  df-xp 5080
This theorem is referenced by:  ex-xp  27147
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