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Theorem xpindir 5841
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 5838 . 2 ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶)) = ((𝐴𝐵) × (𝐶𝐶))
2 inidm 4221 . . 3 (𝐶𝐶) = 𝐶
32xpeq2i 5709 . 2 ((𝐴𝐵) × (𝐶𝐶)) = ((𝐴𝐵) × 𝐶)
41, 3eqtr2i 2757 1 ((𝐴𝐵) × 𝐶) = ((𝐴 × 𝐶) ∩ (𝐵 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1533  cin 3948   × cxp 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-opab 5215  df-xp 5688  df-rel 5689
This theorem is referenced by:  resres  6002  resindi  6005  imainrect  6190  resdmres  6241  txhaus  23571  ustund  24146
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