Theorem iunxiun 5059

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 4956	. . . . . . . 8 ⊢ (𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑥 ∈ 𝐵)
2	1	anbi1i 635	. . . . . . 7 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
3		r19.41v 3195	. . . . . . 7 ⊢ (∃𝑦 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ (∃𝑦 ∈ 𝐴 𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
4	2, 3	bitr4i 281	. . . . . 6 ⊢ ((𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ ∃𝑦 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
5	4	exbii 1871	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ ∃𝑥∃𝑦 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
6		rexcom4 3292	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ ∃𝑥∃𝑦 ∈ 𝐴 (𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
7	5, 6	bitr4i 281	. . . 4 ⊢ (∃𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ 𝐶) ↔ ∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
8		df-rex 3090	. . . 4 ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝑧 ∈ 𝐶 ↔ ∃𝑥(𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵 ∧ 𝑧 ∈ 𝐶))
9		eliun 4956	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶)
10		df-rex 3090	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐵 𝑧 ∈ 𝐶 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
11	9, 10	bitri 278	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
12	11	rexbii 3112	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑦 ∈ 𝐴 ∃𝑥(𝑥 ∈ 𝐵 ∧ 𝑧 ∈ 𝐶))
13	7, 8, 12	3bitr4i 306	. . 3 ⊢ (∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝑧 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)
14		eliun 4956	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝐶 ↔ ∃𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝑧 ∈ 𝐶)
15		eliun 4956	. . 3 ⊢ (𝑧 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶 ↔ ∃𝑦 ∈ 𝐴 𝑧 ∈ ∪ 𝑥 ∈ 𝐵 𝐶)
16	13, 14, 15	3bitr4i 306	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝐶 ↔ 𝑧 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶)
17	16	eqriv 2762	1 ⊢ ∪ 𝑥 ∈ ∪ 𝑦 ∈ 𝐴 𝐵𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑥 ∈ 𝐵 𝐶

Syntax hints:   ∧ wa 400   = wceq 1563  ∃wex 1802   ∈ wcel 2145  ∃wrex 3089  ∪ ciun 4952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-11 2194  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rex 3090  df-v 3459  df-iun 4954

This theorem is referenced by:  ttciun  36887