Theorem uniin2 32622

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
uniin2	⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝐵)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem uniin2

Step	Hyp	Ref	Expression
1		iunin2 5013	. 2 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥)
2		uniiun 5001	. . 3 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
3	2	ineq2i 4157	. 2 ⊢ (𝐴 ∩ ∪ 𝐵) = (𝐴 ∩ ∪ 𝑥 ∈ 𝐵 𝑥)
4	1, 3	eqtr4i 2762	1 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∩ 𝑥) = (𝐴 ∩ ∪ 𝐵)

Syntax hints:   = wceq 1542   ∩ cin 3888  ∪ cuni 4850  ∪ ciun 4933

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rex 3062  df-rab 3390  df-v 3431  df-in 3896  df-uni 4851  df-iun 4935

This theorem is referenced by:  ssdifidllem  33516  ldgenpisyslem1  34307