Theorem iinuni 5027

Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iinuni

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.32v 3270	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝑥) ↔ (𝑦 ∈ 𝐴 ∨ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥))
2		elun 4083	. . . . 5 ⊢ (𝑦 ∈ (𝐴 ∪ 𝑥) ↔ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝑥))
3	2	ralbii 3092	. . . 4 ⊢ (∀𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∪ 𝑥) ↔ ∀𝑥 ∈ 𝐵 (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝑥))
4		vex 3436	. . . . . 6 ⊢ 𝑦 ∈ V
5	4	elint2 4886	. . . . 5 ⊢ (𝑦 ∈ ∩ 𝐵 ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥)
6	5	orbi2i 910	. . . 4 ⊢ ((𝑦 ∈ 𝐴 ∨ 𝑦 ∈ ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∨ ∀𝑥 ∈ 𝐵 𝑦 ∈ 𝑥))
7	1, 3, 6	3bitr4ri 304	. . 3 ⊢ ((𝑦 ∈ 𝐴 ∨ 𝑦 ∈ ∩ 𝐵) ↔ ∀𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∪ 𝑥))
8	7	abbii 2808	. 2 ⊢ {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ ∩ 𝐵)} = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∪ 𝑥)}
9		df-un 3892	. 2 ⊢ (𝐴 ∪ ∩ 𝐵) = {𝑦 ∣ (𝑦 ∈ 𝐴 ∨ 𝑦 ∈ ∩ 𝐵)}
10		df-iin 4927	. 2 ⊢ ∩ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = {𝑦 ∣ ∀𝑥 ∈ 𝐵 𝑦 ∈ (𝐴 ∪ 𝑥)}
11	8, 9, 10	3eqtr4i 2776	1 ⊢ (𝐴 ∪ ∩ 𝐵) = ∩ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥)

Syntax hints:   ∨ wo 844   = wceq 1539   ∈ wcel 2106  {cab 2715  ∀wral 3064   ∪ cun 3885  ∩ cint 4879  ∩ ciin 4925

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-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-v 3434  df-un 3892  df-int 4880  df-iin 4927

This theorem is referenced by: (None)