Theorem iunpw 7789

Description: An indexed union of a power class in terms of the power class of the union of its index. Part of Exercise 24(b) of [Enderton] p. 33. (Contributed by NM, 29-Nov-2003.)

Hypothesis

Ref	Expression
iunpw.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
iunpw	⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 ↔ 𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥)

Distinct variable group:   𝑥,𝐴

Proof of Theorem iunpw

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sseq2 4021	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝐴 → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ ∪ 𝐴))
2	1	biimprcd 250	. . . . . . 7 ⊢ (𝑦 ⊆ ∪ 𝐴 → (𝑥 = ∪ 𝐴 → 𝑦 ⊆ 𝑥))
3	2	reximdv 3167	. . . . . 6 ⊢ (𝑦 ⊆ ∪ 𝐴 → (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 → ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
4	3	com12 32	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 → (𝑦 ⊆ ∪ 𝐴 → ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
5		ssiun 5050	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥)
6		uniiun 5062	. . . . . 6 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
7	5, 6	sseqtrrdi 4046	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥 → 𝑦 ⊆ ∪ 𝐴)
8	4, 7	impbid1 225	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 → (𝑦 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
9		velpw 4609	. . . 4 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝐴)
10		eliun 4999	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
11		velpw 4609	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 𝑥 ↔ 𝑦 ⊆ 𝑥)
12	11	rexbii 3091	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
13	10, 12	bitri 275	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
14	8, 9, 13	3bitr4g 314	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 → (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥))
15	14	eqrdv 2732	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 → 𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥)
16		ssid 4017	. . . . 5 ⊢ ∪ 𝐴 ⊆ ∪ 𝐴
17		iunpw.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
18	17	uniex 7759	. . . . . . 7 ⊢ ∪ 𝐴 ∈ V
19	18	elpw 4608	. . . . . 6 ⊢ (∪ 𝐴 ∈ 𝒫 ∪ 𝐴 ↔ ∪ 𝐴 ⊆ ∪ 𝐴)
20		eleq2 2827	. . . . . 6 ⊢ (𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → (∪ 𝐴 ∈ 𝒫 ∪ 𝐴 ↔ ∪ 𝐴 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥))
21	19, 20	bitr3id 285	. . . . 5 ⊢ (𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → (∪ 𝐴 ⊆ ∪ 𝐴 ↔ ∪ 𝐴 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥))
22	16, 21	mpbii 233	. . . 4 ⊢ (𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → ∪ 𝐴 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥)
23		eliun 4999	. . . 4 ⊢ (∪ 𝐴 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 ∪ 𝐴 ∈ 𝒫 𝑥)
24	22, 23	sylib 218	. . 3 ⊢ (𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → ∃𝑥 ∈ 𝐴 ∪ 𝐴 ∈ 𝒫 𝑥)
25		elssuni 4941	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
26		elpwi 4611	. . . . . . 7 ⊢ (∪ 𝐴 ∈ 𝒫 𝑥 → ∪ 𝐴 ⊆ 𝑥)
27	25, 26	anim12i 613	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ ∪ 𝐴 ∈ 𝒫 𝑥) → (𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥))
28		eqss 4010	. . . . . 6 ⊢ (𝑥 = ∪ 𝐴 ↔ (𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥))
29	27, 28	sylibr 234	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ∪ 𝐴 ∈ 𝒫 𝑥) → 𝑥 = ∪ 𝐴)
30	29	ex 412	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∪ 𝐴 ∈ 𝒫 𝑥 → 𝑥 = ∪ 𝐴))
31	30	reximia 3078	. . 3 ⊢ (∃𝑥 ∈ 𝐴 ∪ 𝐴 ∈ 𝒫 𝑥 → ∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴)
32	24, 31	syl 17	. 2 ⊢ (𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → ∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴)
33	15, 32	impbii 209	1 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 ↔ 𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∃wrex 3067  Vcvv 3477   ⊆ wss 3962  𝒫 cpw 4604  ∪ cuni 4911  ∪ ciun 4995

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-v 3479  df-ss 3979  df-pw 4606  df-uni 4912  df-iun 4997

This theorem is referenced by: (None)