Theorem iunxiun 3947

Description: Separate an indexed union in the index of an indexed union. (Contributed by Mario Carneiro, 5-Dec-2016.)

Assertion

Ref	Expression
iunxiun	⊢ ∪ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶

Distinct variable groups:   𝑥,𝑦   𝑥,𝐴   𝑦,𝐶

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥,𝑦)   𝐶(𝑥)

Proof of Theorem iunxiun

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 3870	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝐵)
2	1	anbi1i 454	. . . . . . 7 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
3		r19.41v 2622	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
4	2, 3	bitr4i 186	. . . . . 6 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ ∃𝑦 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
5	4	exbii 1593	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ ∃𝑥∃𝑦 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
6		rexcom4 2749	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ ∃𝑥∃𝑦 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
7	5, 6	bitr4i 186	. . . 4 ⊢ (∃𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ ∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
8		df-rex 2450	. . . 4 ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝑧 ∈ 𝐶 ↔ ∃𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ 𝐶))
9		eliun 3870	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶)
10		df-rex 2450	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
11	9, 10	bitri 183	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
12	11	rexbii 2473	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
13	7, 8, 12	3bitr4i 211	. . 3 ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝑧 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)
14		eliun 3870	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝐶 ↔ ∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝑧 ∈ 𝐶)
15		eliun 3870	. . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)
16	13, 14, 15	3bitr4i 211	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝐶 ↔ 𝑧 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶)
17	16	eqriv 2162	1 ⊢ ∪ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶

Syntax hints:   ∧ wa 103   = wceq 1343  ∃wex 1480   ∈ wcel 2136  ∃wrex 2445  ∪ ciun 3866

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-iun 3868

This theorem is referenced by: (None)