Theorem iuncom4 3977

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 2516	. . . . . . 7 ⊢ (∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
2	1	rexbii 2539	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
3		rexcom4 2826	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧(𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
4	2, 3	bitri 184	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
5		r19.41v 2689	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
6	5	exbii 1653	. . . . 5 ⊢ (∃𝑧∃𝑥 ∈ 𝐴 (𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
7	4, 6	bitri 184	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧 ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
8		eluni2 3897	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧)
9	8	rexbii 2539	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑦 ∈ 𝑧)
10		df-rex 2516	. . . . 5 ⊢ (∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧 ↔ ∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧))
11		eliun 3974	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
12	11	anbi1i 458	. . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
13	12	exbii 1653	. . . . 5 ⊢ (∃𝑧(𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ∧ 𝑦 ∈ 𝑧) ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
14	10, 13	bitri 184	. . . 4 ⊢ (∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧 ↔ ∃𝑧(∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ∧ 𝑦 ∈ 𝑧))
15	7, 9, 14	3bitr4i 212	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵 ↔ ∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧)
16		eliun 3974	. . 3 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ ∪ 𝐵)
17		eluni2 3897	. . 3 ⊢ (𝑦 ∈ ∪ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵𝑦 ∈ 𝑧)
18	15, 16, 17	3bitr4i 212	. 2 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 ↔ 𝑦 ∈ ∪ ∪ 𝑥 ∈ 𝐴 𝐵)
19	18	eqriv 2228	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∪ 𝐵 = ∪ ∪ 𝑥 ∈ 𝐴 𝐵

Syntax hints:   ∧ wa 104   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∃wrex 2511  ∪ cuni 3893  ∪ ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-uni 3894  df-iun 3972

This theorem is referenced by:  tgidm  14797