Theorem xpindi 5742

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 5741	. 2 ⊢ ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶)) = ((𝐴 ∩ 𝐴) × (𝐵 ∩ 𝐶))
2		inidm 4152	. . 3 ⊢ (𝐴 ∩ 𝐴) = 𝐴
3	2	xpeq1i 5615	. 2 ⊢ ((𝐴 ∩ 𝐴) × (𝐵 ∩ 𝐶)) = (𝐴 × (𝐵 ∩ 𝐶))
4	1, 3	eqtr2i 2767	1 ⊢ (𝐴 × (𝐵 ∩ 𝐶)) = ((𝐴 × 𝐵) ∩ (𝐴 × 𝐶))

Syntax hints:   = wceq 1539   ∩ cin 3886   × cxp 5587

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-opab 5137  df-xp 5595  df-rel 5596

This theorem is referenced by:  xpriindi  5745  djuassen  9934  xpdjuen  9935  fpwwe2lem12  10398  txhaus  22798  ustund  23373