Theorem iuncom4 4528

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 2918	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
2	1	rexbii 3041	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
3		rexcom4 3225	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
4	2, 3	bitri 264	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
5		r19.41v 3089	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
6	5	exbii 1774	. . . . 5 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
7	4, 6	bitri 264	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
8		eluni2 4440	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧)
9	8	rexbii 3041	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧)
10		df-rex 2918	. . . . 5 ⊢ (∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧))
11		eliun 4524	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
12	11	anbi1i 731	. . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
13	12	exbii 1774	. . . . 5 ⊢ (∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
14	10, 13	bitri 264	. . . 4 ⊢ (∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧 ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
15	7, 9, 14	3bitr4i 292	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧)
16		eliun 4524	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵)
17		eluni2 4440	. . 3 ⊢ (𝑦 ∈ ∪ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧)
18	15, 16, 17	3bitr4i 292	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 ↔ 𝑦 ∈ ∪ ∪ 𝑥 ∈ 𝐴 𝐵)
19	18	eqriv 2619	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 = ∪ ∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 384   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∃wrex 2913  ∪ cuni 4436  ∪ ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-v 3202  df-uni 4437  df-iun 4522

This theorem is referenced by:  ituniiun  9244  tgidm  20784  txcmplem2  21445