Theorem iuncom4 3828

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 2423	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
2	1	rexbii 2445	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
3		rexcom4 2712	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
4	2, 3	bitri 183	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
5		r19.41v 2590	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
6	5	exbii 1585	. . . . 5 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
7	4, 6	bitri 183	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
8		eluni2 3748	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧)
9	8	rexbii 2445	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧)
10		df-rex 2423	. . . . 5 ⊢ (∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧))
11		eliun 3825	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
12	11	anbi1i 454	. . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
13	12	exbii 1585	. . . . 5 ⊢ (∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
14	10, 13	bitri 183	. . . 4 ⊢ (∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧 ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
15	7, 9, 14	3bitr4i 211	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧)
16		eliun 3825	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵)
17		eluni2 3748	. . 3 ⊢ (𝑦 ∈ ∪ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧)
18	15, 16, 17	3bitr4i 211	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 ↔ 𝑦 ∈ ∪ ∪ 𝑥 ∈ 𝐴 𝐵)
19	18	eqriv 2137	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 = ∪ ∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∃wrex 2418  ∪ cuni 3744  ∪ ciun 3821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-uni 3745  df-iun 3823

This theorem is referenced by:  tgidm  12282