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Theorem xpindi 5160
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 5159 . 2 ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶)) = ((𝐴𝐴) × (𝐵𝐶))
2 inidm 3778 . . 3 (𝐴𝐴) = 𝐴
32xpeq1i 5044 . 2 ((𝐴𝐴) × (𝐵𝐶)) = (𝐴 × (𝐵𝐶))
41, 3eqtr2i 2627 1 (𝐴 × (𝐵𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:   = wceq 1474  cin 3533   × cxp 5021
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pr 4823
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ral 2895  df-rex 2896  df-rab 2899  df-v 3169  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-nul 3869  df-if 4031  df-sn 4120  df-pr 4122  df-op 4126  df-opab 4633  df-xp 5029  df-rel 5030
This theorem is referenced by:  xpriindi  5163  xpcdaen  8860  fpwwe2lem13  9315  txhaus  21197  ustund  21772
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