Theorem iuncom4 4964

Description: Commutation of union with indexed union. (Contributed by Mario Carneiro, 18-Jan-2014.)

Assertion

Ref	Expression
iuncom4	⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 = ∪ ∪ 𝑥 ∈ 𝐴 𝐵

Proof of Theorem iuncom4

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-rex 3054	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
2	1	rexbii 3076	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
3		rexcom4 3264	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
4	2, 3	bitri 275	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
5		r19.41v 3167	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
6	5	exbii 1848	. . . . 5 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
7	4, 6	bitri 275	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
8		eluni2 4875	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧)
9	8	rexbii 3076	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧)
10		df-rex 3054	. . . . 5 ⊢ (∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧))
11		eliun 4959	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
12	11	anbi1i 624	. . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
13	12	exbii 1848	. . . . 5 ⊢ (∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
14	10, 13	bitri 275	. . . 4 ⊢ (∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧 ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
15	7, 9, 14	3bitr4i 303	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧)
16		eliun 4959	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵)
17		eluni2 4875	. . 3 ⊢ (𝑦 ∈ ∪ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧)
18	15, 16, 17	3bitr4i 303	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 ↔ 𝑦 ∈ ∪ ∪ 𝑥 ∈ 𝐴 𝐵)
19	18	eqriv 2726	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 = ∪ ∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃wrex 3053  ∪ cuni 4871  ∪ ciun 4955

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-11 2158  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rex 3054  df-v 3449  df-uni 4872  df-iun 4957

This theorem is referenced by:  ituniiun  10375  tgidm  22867  txcmplem2  23529