Theorem uniin1 5032

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 5029	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝑥 ∩ 𝐵)
2		uniiun 5016	. . 3 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
3	2	ineq1i 4113	. 2 ⊢ (∪ 𝐴 ∩ 𝐵) = (∪ 𝑥 ∈ 𝐴 𝑥 ∩ 𝐵)
4	1, 3	eqtr4i 2786	1 ⊢ ∪ 𝑥 ∈ 𝐴 (𝑥 ∩ 𝐵) = (∪ 𝐴 ∩ 𝐵)

This proof depends on syntax axioms:   = wceq 1569   ∩ cin 4089  ∪ cuni 4866  ∪ ciun 4950

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-in 4091  df-uni 4867  df-iun 4952

This theorem is used by: (None)