Theorem uniuni 7782

Description: Expression for double union that moves union into a class abstraction. (Contributed by FL, 28-May-2007.)

Assertion

Ref	Expression
uniuni	⊢ ∪ ∪ 𝐴 = ∪ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}

Distinct variable group:   𝑥,𝐴,𝑦

Proof of Theorem uniuni

Dummy variables 𝑣 𝑧 𝑢 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eluni 4910	. . . . . 6 ⊢ (𝑢 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
2	1	anbi2i 623	. . . . 5 ⊢ ((𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴) ↔ (𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
3	2	exbii 1848	. . . 4 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴) ↔ ∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
4		19.42v 1953	. . . . . . 7 ⊢ (∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ (𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
5	4	bicomi 224	. . . . . 6 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
6	5	exbii 1848	. . . . 5 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑢∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
7		excom 2162	. . . . . 6 ⊢ (∃𝑢∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦∃𝑢(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
8		anass 468	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦) ∧ 𝑦 ∈ 𝐴) ↔ (𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
9		ancom 460	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦) ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)))
10	8, 9	bitr3i 277	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)))
11	10	2exbii 1849	. . . . . 6 ⊢ (∃𝑦∃𝑢(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦∃𝑢(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)))
12		exdistr 1954	. . . . . 6 ⊢ (∃𝑦∃𝑢(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)))
13	7, 11, 12	3bitri 297	. . . . 5 ⊢ (∃𝑢∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)))
14		eluni 4910	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝑦 ↔ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦))
15	14	bicomi 224	. . . . . . 7 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦) ↔ 𝑧 ∈ ∪ 𝑦)
16	15	anbi2i 623	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦))
17	16	exbii 1848	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦))
18	6, 13, 17	3bitri 297	. . . 4 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦))
19		vuniex 7759	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ V
20		eleq2 2830	. . . . . . . . . 10 ⊢ (𝑣 = ∪ 𝑦 → (𝑧 ∈ 𝑣 ↔ 𝑧 ∈ ∪ 𝑦))
21	19, 20	ceqsexv 3532	. . . . . . . . 9 ⊢ (∃𝑣(𝑣 = ∪ 𝑦 ∧ 𝑧 ∈ 𝑣) ↔ 𝑧 ∈ ∪ 𝑦)
22		exancom 1861	. . . . . . . . 9 ⊢ (∃𝑣(𝑣 = ∪ 𝑦 ∧ 𝑧 ∈ 𝑣) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦))
23	21, 22	bitr3i 277	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝑦 ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦))
24	23	anbi2i 623	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)))
25		19.42v 1953	. . . . . . 7 ⊢ (∃𝑣(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)))
26		ancom 460	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) ↔ ((𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦) ∧ 𝑦 ∈ 𝐴))
27		anass 468	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦) ∧ 𝑦 ∈ 𝐴) ↔ (𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
28	26, 27	bitri 275	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) ↔ (𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
29	28	exbii 1848	. . . . . . 7 ⊢ (∃𝑣(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
30	24, 25, 29	3bitr2i 299	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
31	30	exbii 1848	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦) ↔ ∃𝑦∃𝑣(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
32		excom 2162	. . . . 5 ⊢ (∃𝑦∃𝑣(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑣∃𝑦(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
33		exdistr 1954	. . . . . 6 ⊢ (∃𝑣∃𝑦(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
34		vex 3484	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
35		eqeq1 2741	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑣 → (𝑥 = ∪ 𝑦 ↔ 𝑣 = ∪ 𝑦))
36	35	anbi1d 631	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → ((𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
37	36	exbidv 1921	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
38	34, 37	elab 3679	. . . . . . . . 9 ⊢ (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)} ↔ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴))
39	38	bicomi 224	. . . . . . . 8 ⊢ (∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)})
40	39	anbi2i 623	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑣 ∧ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ (𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}))
41	40	exbii 1848	. . . . . 6 ⊢ (∃𝑣(𝑧 ∈ 𝑣 ∧ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}))
42	33, 41	bitri 275	. . . . 5 ⊢ (∃𝑣∃𝑦(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}))
43	31, 32, 42	3bitri 297	. . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}))
44	3, 18, 43	3bitri 297	. . 3 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}))
45	44	abbii 2809	. 2 ⊢ {𝑧 ∣ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴)} = {𝑧 ∣ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)})}
46		df-uni 4908	. 2 ⊢ ∪ ∪ 𝐴 = {𝑧 ∣ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴)}
47		df-uni 4908	. 2 ⊢ ∪ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)} = {𝑧 ∣ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)})}
48	45, 46, 47	3eqtr4i 2775	1 ⊢ ∪ ∪ 𝐴 = ∪ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714  ∪ cuni 4907

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2157  ax-ext 2708  ax-sep 5296  ax-un 7755

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-uni 4908

This theorem is referenced by: (None)