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 4161	. 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 1570   ∩ cin 3897  ∪ cuni 4866  ∪ ciun 4950

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-in 3905  df-ss 3915  df-uni 4867  df-iun 4952

This theorem is used by: (None)