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Theorem xpindi 5735
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 5734 . 2 ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶)) = ((𝐴𝐴) × (𝐵𝐶))
2 inidm 4152 . . 3 (𝐴𝐴) = 𝐴
32xpeq1i 5610 . 2 ((𝐴𝐴) × (𝐵𝐶)) = (𝐴 × (𝐵𝐶))
41, 3eqtr2i 2767 1 (𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1539  cin 3885   × cxp 5582
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5221  ax-nul 5228  ax-pr 5350
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3431  df-dif 3889  df-un 3891  df-in 3893  df-ss 3903  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5136  df-xp 5590  df-rel 5591
This theorem is referenced by:  xpriindi  5738  djuassen  9944  xpdjuen  9945  fpwwe2lem12  10408  txhaus  22808  ustund  23383
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