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Theorem xpindi 5794
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 5793 . 2 ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶)) = ((𝐴𝐴) × (𝐵𝐶))
2 inidm 4183 . . 3 (𝐴𝐴) = 𝐴
32xpeq1i 5664 . 2 ((𝐴𝐴) × (𝐵𝐶)) = (𝐴 × (𝐵𝐶))
41, 3eqtr2i 2766 1 (𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  cin 3914   × cxp 5636
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-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-opab 5173  df-xp 5644  df-rel 5645
This theorem is referenced by:  xpriindi  5797  djuassen  10121  xpdjuen  10122  fpwwe2lem12  10585  txhaus  23014  ustund  23589
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