Theorem iunpw 7473

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 3941	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝐴 → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ ∪ 𝐴))
2	1	biimprcd 253	. . . . . . 7 ⊢ (𝑦 ⊆ ∪ 𝐴 → (𝑥 = ∪ 𝐴 → 𝑦 ⊆ 𝑥))
3	2	reximdv 3232	. . . . . 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 4933	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥)
6		uniiun 4945	. . . . . 6 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
7	5, 6	sseqtrrdi 3966	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥 → 𝑦 ⊆ ∪ 𝐴)
8	4, 7	impbid1 228	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 → (𝑦 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
9		velpw 4502	. . . 4 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝐴)
10		eliun 4885	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
11		velpw 4502	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 𝑥 ↔ 𝑦 ⊆ 𝑥)
12	11	rexbii 3210	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
13	10, 12	bitri 278	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
14	8, 9, 13	3bitr4g 317	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 → (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥))
15	14	eqrdv 2796	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 → 𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥)
16		ssid 3937	. . . . 5 ⊢ ∪ 𝐴 ⊆ ∪ 𝐴
17		iunpw.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
18	17	uniex 7447	. . . . . . 7 ⊢ ∪ 𝐴 ∈ V
19	18	elpw 4501	. . . . . 6 ⊢ (∪ 𝐴 ∈ 𝒫 ∪ 𝐴 ↔ ∪ 𝐴 ⊆ ∪ 𝐴)
20		eleq2 2878	. . . . . 6 ⊢ (𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → (∪ 𝐴 ∈ 𝒫 ∪ 𝐴 ↔ ∪ 𝐴 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥))
21	19, 20	bitr3id 288	. . . . 5 ⊢ (𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → (∪ 𝐴 ⊆ ∪ 𝐴 ↔ ∪ 𝐴 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥))
22	16, 21	mpbii 236	. . . 4 ⊢ (𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → ∪ 𝐴 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥)
23		eliun 4885	. . . 4 ⊢ (∪ 𝐴 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 ∪ 𝐴 ∈ 𝒫 𝑥)
24	22, 23	sylib 221	. . 3 ⊢ (𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → ∃𝑥 ∈ 𝐴 ∪ 𝐴 ∈ 𝒫 𝑥)
25		elssuni 4830	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
26		elpwi 4506	. . . . . . 7 ⊢ (∪ 𝐴 ∈ 𝒫 𝑥 → ∪ 𝐴 ⊆ 𝑥)
27	25, 26	anim12i 615	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ ∪ 𝐴 ∈ 𝒫 𝑥) → (𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥))
28		eqss 3930	. . . . . 6 ⊢ (𝑥 = ∪ 𝐴 ↔ (𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥))
29	27, 28	sylibr 237	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ∪ 𝐴 ∈ 𝒫 𝑥) → 𝑥 = ∪ 𝐴)
30	29	ex 416	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∪ 𝐴 ∈ 𝒫 𝑥 → 𝑥 = ∪ 𝐴))
31	30	reximia 3205	. . 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 212	1 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 ↔ 𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥)

Syntax hints:   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  Vcvv 3441   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800  ∪ ciun 4881

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-in 3888  df-ss 3898  df-pw 4499  df-uni 4801  df-iun 4883

This theorem is referenced by: (None)