Theorem uniuni 4330

Description: Expression for double union that moves union into a class builder. (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 3703	. . . . . 6 ⊢ (𝑢 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
2	1	anbi2i 450	. . . . 5 ⊢ ((𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴) ↔ (𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
3	2	exbii 1565	. . . 4 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴) ↔ ∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
4		19.42v 1858	. . . . . . 7 ⊢ (∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ (𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
5	4	bicomi 131	. . . . . 6 ⊢ ((𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
6	5	exbii 1565	. . . . 5 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑢∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
7		excom 1623	. . . . . 6 ⊢ (∃𝑢∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦∃𝑢(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
8		anass 396	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦) ∧ 𝑦 ∈ 𝐴) ↔ (𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)))
9		ancom 264	. . . . . . . 8 ⊢ (((𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦) ∧ 𝑦 ∈ 𝐴) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)))
10	8, 9	bitr3i 185	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ (𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)))
11	10	2exbii 1566	. . . . . 6 ⊢ (∃𝑦∃𝑢(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦∃𝑢(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)))
12		exdistr 1859	. . . . . 6 ⊢ (∃𝑦∃𝑢(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)))
13	7, 11, 12	3bitri 205	. . . . 5 ⊢ (∃𝑢∃𝑦(𝑧 ∈ 𝑢 ∧ (𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)))
14		eluni 3703	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝑦 ↔ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦))
15	14	bicomi 131	. . . . . . 7 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦) ↔ 𝑧 ∈ ∪ 𝑦)
16	15	anbi2i 450	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)) ↔ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦))
17	16	exbii 1565	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ 𝑦)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦))
18	6, 13, 17	3bitri 205	. . . 4 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ ∃𝑦(𝑢 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦))
19		vex 2658	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
20	19	uniex 4317	. . . . . . . . . 10 ⊢ ∪ 𝑦 ∈ V
21		eleq2 2176	. . . . . . . . . 10 ⊢ (𝑣 = ∪ 𝑦 → (𝑧 ∈ 𝑣 ↔ 𝑧 ∈ ∪ 𝑦))
22	20, 21	ceqsexv 2694	. . . . . . . . 9 ⊢ (∃𝑣(𝑣 = ∪ 𝑦 ∧ 𝑧 ∈ 𝑣) ↔ 𝑧 ∈ ∪ 𝑦)
23		exancom 1568	. . . . . . . . 9 ⊢ (∃𝑣(𝑣 = ∪ 𝑦 ∧ 𝑧 ∈ 𝑣) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦))
24	22, 23	bitr3i 185	. . . . . . . 8 ⊢ (𝑧 ∈ ∪ 𝑦 ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦))
25	24	anbi2i 450	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)))
26		19.42v 1858	. . . . . . 7 ⊢ (∃𝑣(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) ↔ (𝑦 ∈ 𝐴 ∧ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)))
27		ancom 264	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) ↔ ((𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦) ∧ 𝑦 ∈ 𝐴))
28		anass 396	. . . . . . . . 9 ⊢ (((𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦) ∧ 𝑦 ∈ 𝐴) ↔ (𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
29	27, 28	bitri 183	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) ↔ (𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
30	29	exbii 1565	. . . . . . 7 ⊢ (∃𝑣(𝑦 ∈ 𝐴 ∧ (𝑧 ∈ 𝑣 ∧ 𝑣 = ∪ 𝑦)) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
31	25, 26, 30	3bitr2i 207	. . . . . 6 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
32	31	exbii 1565	. . . . 5 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦) ↔ ∃𝑦∃𝑣(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
33		excom 1623	. . . . 5 ⊢ (∃𝑦∃𝑣(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑣∃𝑦(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
34		exdistr 1859	. . . . . 6 ⊢ (∃𝑣∃𝑦(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
35		vex 2658	. . . . . . . . . 10 ⊢ 𝑣 ∈ V
36		eqeq1 2119	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑣 → (𝑥 = ∪ 𝑦 ↔ 𝑣 = ∪ 𝑦))
37	36	anbi1d 458	. . . . . . . . . . 11 ⊢ (𝑥 = 𝑣 → ((𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
38	37	exbidv 1777	. . . . . . . . . 10 ⊢ (𝑥 = 𝑣 → (∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)))
39	35, 38	elab 2796	. . . . . . . . 9 ⊢ (𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)} ↔ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴))
40	39	bicomi 131	. . . . . . . 8 ⊢ (∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴) ↔ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)})
41	40	anbi2i 450	. . . . . . 7 ⊢ ((𝑧 ∈ 𝑣 ∧ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ (𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}))
42	41	exbii 1565	. . . . . 6 ⊢ (∃𝑣(𝑧 ∈ 𝑣 ∧ ∃𝑦(𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}))
43	34, 42	bitri 183	. . . . 5 ⊢ (∃𝑣∃𝑦(𝑧 ∈ 𝑣 ∧ (𝑣 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}))
44	32, 33, 43	3bitri 205	. . . 4 ⊢ (∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑧 ∈ ∪ 𝑦) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}))
45	3, 18, 44	3bitri 205	. . 3 ⊢ (∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴) ↔ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}))
46	45	abbii 2228	. 2 ⊢ {𝑧 ∣ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴)} = {𝑧 ∣ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)})}
47		df-uni 3701	. 2 ⊢ ∪ ∪ 𝐴 = {𝑧 ∣ ∃𝑢(𝑧 ∈ 𝑢 ∧ 𝑢 ∈ ∪ 𝐴)}
48		df-uni 3701	. 2 ⊢ ∪ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)} = {𝑧 ∣ ∃𝑣(𝑧 ∈ 𝑣 ∧ 𝑣 ∈ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)})}
49	46, 47, 48	3eqtr4i 2143	1 ⊢ ∪ ∪ 𝐴 = ∪ {𝑥 ∣ ∃𝑦(𝑥 = ∪ 𝑦 ∧ 𝑦 ∈ 𝐴)}

Syntax hints:   ∧ wa 103   = wceq 1312  ∃wex 1449   ∈ wcel 1461  {cab 2099  ∪ cuni 3700

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1404  ax-7 1405  ax-gen 1406  ax-ie1 1450  ax-ie2 1451  ax-8 1463  ax-10 1464  ax-11 1465  ax-i12 1466  ax-bndl 1467  ax-4 1468  ax-13 1472  ax-14 1473  ax-17 1487  ax-i9 1491  ax-ial 1495  ax-i5r 1496  ax-ext 2095  ax-sep 4004  ax-un 4313

This theorem depends on definitions:  df-bi 116  df-tru 1315  df-nf 1418  df-sb 1717  df-clab 2100  df-cleq 2106  df-clel 2109  df-nfc 2242  df-rex 2394  df-v 2657  df-uni 3701

This theorem is referenced by: (None)