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Theorem xpindi 5288
Description: Distributive law for Cartesian product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
Assertion
Ref Expression
xpindi (𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))

Proof of Theorem xpindi
StepHypRef Expression
1 inxp 5287 . 2 ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶)) = ((𝐴𝐴) × (𝐵𝐶))
2 inidm 3855 . . 3 (𝐴𝐴) = 𝐴
32xpeq1i 5169 . 2 ((𝐴𝐴) × (𝐵𝐶)) = (𝐴 × (𝐵𝐶))
41, 3eqtr2i 2674 1 (𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1523  cin 3606   × 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  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  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-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-xp 5149  df-rel 5150
This theorem is referenced by:  xpriindi  5291  xpcdaen  9043  fpwwe2lem13  9502  txhaus  21498  ustund  22072
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