Theorem iuncom4 4956

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 3062	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
2	1	rexbii 3084	. . . . . 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 1850	. . . . 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 4868	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧)
9	8	rexbii 3084	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧)
10		df-rex 3062	. . . . 5 ⊢ (∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧))
11		eliun 4951	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
12	11	anbi1i 625	. . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
13	12	exbii 1850	. . . . 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 4951	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵)
17		eluni2 4868	. . 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 2734	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 = ∪ ∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∃wrex 3061  ∪ cuni 4864  ∪ ciun 4947

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-11 2163  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-rex 3062  df-v 3443  df-uni 4865  df-iun 4949

This theorem is referenced by:  ituniiun  10336  tgidm  22928  txcmplem2  23590