Theorem iunpw 7493

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 3993	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝐴 → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ ∪ 𝐴))
2	1	biimprcd 252	. . . . . . 7 ⊢ (𝑦 ⊆ ∪ 𝐴 → (𝑥 = ∪ 𝐴 → 𝑦 ⊆ 𝑥))
3	2	reximdv 3273	. . . . . 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 4970	. . . . . 6 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥 → 𝑦 ⊆ ∪ 𝑥 ∈ 𝐴 𝑥)
6		uniiun 4982	. . . . . 6 ⊢ ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝑥
7	5, 6	sseqtrrdi 4018	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥 → 𝑦 ⊆ ∪ 𝐴)
8	4, 7	impbid1 227	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 → (𝑦 ⊆ ∪ 𝐴 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥))
9		velpw 4544	. . . 4 ⊢ (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ⊆ ∪ 𝐴)
10		eliun 4923	. . . . 5 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥)
11		velpw 4544	. . . . . 6 ⊢ (𝑦 ∈ 𝒫 𝑥 ↔ 𝑦 ⊆ 𝑥)
12	11	rexbii 3247	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑦 ∈ 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
13	10, 12	bitri 277	. . . 4 ⊢ (𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 𝑦 ⊆ 𝑥)
14	8, 9, 13	3bitr4g 316	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 → (𝑦 ∈ 𝒫 ∪ 𝐴 ↔ 𝑦 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥))
15	14	eqrdv 2819	. 2 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 → 𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥)
16		ssid 3989	. . . . 5 ⊢ ∪ 𝐴 ⊆ ∪ 𝐴
17		iunpw.1	. . . . . . . 8 ⊢ 𝐴 ∈ V
18	17	uniex 7467	. . . . . . 7 ⊢ ∪ 𝐴 ∈ V
19	18	elpw 4543	. . . . . 6 ⊢ (∪ 𝐴 ∈ 𝒫 ∪ 𝐴 ↔ ∪ 𝐴 ⊆ ∪ 𝐴)
20		eleq2 2901	. . . . . 6 ⊢ (𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → (∪ 𝐴 ∈ 𝒫 ∪ 𝐴 ↔ ∪ 𝐴 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥))
21	19, 20	syl5bbr 287	. . . . 5 ⊢ (𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → (∪ 𝐴 ⊆ ∪ 𝐴 ↔ ∪ 𝐴 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥))
22	16, 21	mpbii 235	. . . 4 ⊢ (𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → ∪ 𝐴 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥)
23		eliun 4923	. . . 4 ⊢ (∪ 𝐴 ∈ ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 ↔ ∃𝑥 ∈ 𝐴 ∪ 𝐴 ∈ 𝒫 𝑥)
24	22, 23	sylib 220	. . 3 ⊢ (𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥 → ∃𝑥 ∈ 𝐴 ∪ 𝐴 ∈ 𝒫 𝑥)
25		elssuni 4868	. . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ⊆ ∪ 𝐴)
26		elpwi 4548	. . . . . . 7 ⊢ (∪ 𝐴 ∈ 𝒫 𝑥 → ∪ 𝐴 ⊆ 𝑥)
27	25, 26	anim12i 614	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ ∪ 𝐴 ∈ 𝒫 𝑥) → (𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥))
28		eqss 3982	. . . . . 6 ⊢ (𝑥 = ∪ 𝐴 ↔ (𝑥 ⊆ ∪ 𝐴 ∧ ∪ 𝐴 ⊆ 𝑥))
29	27, 28	sylibr 236	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ ∪ 𝐴 ∈ 𝒫 𝑥) → 𝑥 = ∪ 𝐴)
30	29	ex 415	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∪ 𝐴 ∈ 𝒫 𝑥 → 𝑥 = ∪ 𝐴))
31	30	reximia 3242	. . 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 211	1 ⊢ (∃𝑥 ∈ 𝐴 𝑥 = ∪ 𝐴 ↔ 𝒫 ∪ 𝐴 = ∪ 𝑥 ∈ 𝐴 𝒫 𝑥)

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∃wrex 3139  Vcvv 3494   ⊆ wss 3936  𝒫 cpw 4539  ∪ cuni 4838  ∪ ciun 4919

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-v 3496  df-in 3943  df-ss 3952  df-pw 4541  df-uni 4839  df-iun 4921

This theorem is referenced by: (None)