Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  xpindi Structured version   Unicode version

Theorem xpindi 5146
 Description: Distributive law for Cartesian product over intersection. Theorem 102 of [Suppes] p. 52. (Contributed by NM, 26-Sep-2004.)
Assertion
Ref Expression
xpindi

Proof of Theorem xpindi
StepHypRef Expression
1 inxp 5145 . 2
2 inidm 3703 . . 3
32xpeq1i 5028 . 2
41, 3eqtr2i 2487 1
 Colors of variables: wff setvar class Syntax hints:   wceq 1395   cin 3470   cxp 5006 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-opab 4516  df-xp 5014  df-rel 5015 This theorem is referenced by:  xpriindi  5149  xpcdaen  8580  fpwwe2lem13  9037  txhaus  20274  ustund  20850
 Copyright terms: Public domain W3C validator