Theorem xpindi 5844

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 × (𝐵 ∩ 𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))

Proof of Theorem xpindi

Step	Hyp	Ref	Expression
1		inxp 5842	. 2 ⊢ ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶)) = ((𝐴 ∩ 𝐴) × (𝐵 ∩ 𝐶))
2		inidm 4227	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	xpeq1i 5711	. 2 ⊢ ((𝐴 ∩ 𝐴) × (𝐵 ∩ 𝐶)) = (𝐴 × (𝐵 ∩ 𝐶))
4	1, 3	eqtr2i 2766	1 ⊢ (𝐴 × (𝐵 ∩ 𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))

Syntax hints:   = wceq 1540   ∩ cin 3950   × cxp 5683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-opab 5206  df-xp 5691  df-rel 5692

This theorem is referenced by:  xpriindi  5847  djuassen  10219  xpdjuen  10220  fpwwe2lem12  10682  txhaus  23655  ustund  24230