Theorem uniin1 30767

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 4997	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝑥 ∩ 𝐵)
2		uniiun 4984	. . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
3	2	ineq1i 4140	. 2 ⊢ (∪ 𝐴 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝑥 ∩ 𝐵)
4	1, 3	eqtr4i 2770	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = (∪ 𝐴 ∩ 𝐵)

Syntax hints:   = wceq 1543   ∩ cin 3883  ∪ cuni 4836  ∪ ciun 4921

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3425  df-in 3891  df-ss 3901  df-uni 4837  df-iun 4923

This theorem is referenced by: (None)