Theorem uniin1 30891

Description: Union of intersection. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. (Contributed by Thierry Arnoux, 21-Jun-2020.)

Assertion

Ref	Expression
uniin1	⊢ ∪ 𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = (∪ 𝐴 ∩ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem uniin1

Step	Hyp	Ref	Expression
1		iunin1 5001	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝑥 ∩ 𝐵)
2		uniiun 4988	. . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
3	2	ineq1i 4142	. 2 ⊢ (∪ 𝐴 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝑥 ∩ 𝐵)
4	1, 3	eqtr4i 2769	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = (∪ 𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1539   ∩ cin 3886  ∪ cuni 4839  ∪ ciun 4924

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-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-in 3894  df-ss 3904  df-uni 4840  df-iun 4926

This theorem is referenced by: (None)