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Theorem xpindir 5848
Description: Distributive law for Cartesian product over intersection. Similar to Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
Assertion
Ref Expression
xpindir ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))

Proof of Theorem xpindir
StepHypRef Expression
1 inxp 5845 . 2 ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) = ((𝐴𝐵) × (𝐶𝐶))
2 inidm 4235 . . 3 (𝐶𝐶) = 𝐶
32xpeq2i 5716 . 2 ((𝐴𝐵) × (𝐶𝐶)) = ((𝐴𝐵) × 𝐶)
41, 3eqtr2i 2764 1 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  cin 3962   × cxp 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-opab 5211  df-xp 5695  df-rel 5696
This theorem is referenced by:  resres  6013  resindi  6016  imainrect  6203  resdmres  6254  txhaus  23671  ustund  24246
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