Theorem iunpw 2040

Description: The power class of an intersection in terms of indexed intersection. Part of Exercise 24(b) of [Enderton] p. 33.

Hypothesis

Ref	Expression
iunpw.1	⊢ A ∈ V

Assertion

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

Distinct variable group(s):   x,A

Proof of Theorem iunpw

Step	Hyp	Ref	Expression
1		sseq2 1522	. . . . . . . 8 ⊢ (x = ∪A → (y ⊆ x ↔ y ⊆ ∪A))
2	1	biimprcd 138	. . . . . . 7 ⊢ (y ⊆ ∪A → (x = ∪A → y ⊆ x))
3	2	r19.22sdv 1279	. . . . . 6 ⊢ (y ⊆ ∪A → (∃x ∈ A x = ∪A → ∃x ∈ A y ⊆ x))
4	3	com12 13	. . . . 5 ⊢ (∃x ∈ A x = ∪A → (y ⊆ ∪A → ∃x ∈ A y ⊆ x))
5		ssiun 2018	. . . . . . 7 ⊢ (∃x ∈ A y ⊆ x → y ⊆ ∪x ∈ A x)
6		uniiun 2026	. . . . . . 7 ⊢ ∪A = ∪x ∈ A x
7	5, 6	syl6ssr 1547	. . . . . 6 ⊢ (∃x ∈ A y ⊆ x → y ⊆ ∪A)
8	7	a1i 7	. . . . 5 ⊢ (∃x ∈ A x = ∪A → (∃x ∈ A y ⊆ x → y ⊆ ∪A))
9	4, 8	impbid 397	. . . 4 ⊢ (∃x ∈ A x = ∪A → (y ⊆ ∪A ↔ ∃x ∈ A y ⊆ x))
10		visset 1350	. . . . 5 ⊢ y ∈ V
11	10	elpw 1801	. . . 4 ⊢ (y ∈ ℘∪A ↔ y ⊆ ∪A)
12		eliun 1998	. . . . 5 ⊢ (y ∈ ∪x ∈ A ℘x ↔ ∃x ∈ A y ∈ ℘x)
13		df-pw 1799	. . . . . . 7 ⊢ ℘x = {y∣y ⊆ x}
14	13	cleqabi 1176	. . . . . 6 ⊢ (y ∈ ℘x ↔ y ⊆ x)
15	14	birex 1224	. . . . 5 ⊢ (∃x ∈ A y ∈ ℘x ↔ ∃x ∈ A y ⊆ x)
16	12, 15	bitr 151	. . . 4 ⊢ (y ∈ ∪x ∈ A ℘x ↔ ∃x ∈ A y ⊆ x)
17	9, 11, 16	3bitr4g 428	. . 3 ⊢ (∃x ∈ A x = ∪A → (y ∈ ℘∪A ↔ y ∈ ∪x ∈ A ℘x))
18	17	cleqrd 1100	. 2 ⊢ (∃x ∈ A x = ∪A → ℘∪A = ∪x ∈ A ℘x)
19		ssid 1519	. . . . 5 ⊢ ∪A ⊆ ∪A
20		eleq2 1150	. . . . . 6 ⊢ (℘∪A = ∪x ∈ A ℘x → (∪A ∈ ℘ ∪A ↔ ∪A ∈ ∪ x ∈ A ℘x))
21		iunpw.1	. . . . . . . 8 ⊢ A ∈ V
22	21	uniex 1947	. . . . . . 7 ⊢ ∪A ∈ V
23	22	elpw 1801	. . . . . 6 ⊢ (∪A ∈ ℘ ∪A ↔ ∪A ⊆ ∪A)
24	20, 23	syl5bbr 412	. . . . 5 ⊢ (℘∪A = ∪x ∈ A ℘x → (∪A ⊆ ∪A ↔ ∪A ∈ ∪ x ∈ A ℘x))
25	19, 24	mpbii 168	. . . 4 ⊢ (℘∪A = ∪x ∈ A ℘x → ∪A ∈ ∪ x ∈ A ℘x)
26		eliun 1998	. . . 4 ⊢ (∪A ∈ ∪ x ∈ A ℘x ↔ ∃x ∈ A ∪A ∈ ℘ x)
27	25, 26	sylib 173	. . 3 ⊢ (℘∪A = ∪x ∈ A ℘x → ∃x ∈ A ∪A ∈ ℘ x)
28		elssuni 1940	. . . . . . 7 ⊢ (x ∈ A → x ⊆ ∪A)
29	22	elpw 1801	. . . . . . . 8 ⊢ (∪A ∈ ℘ x ↔ ∪A ⊆ x)
30	29	biimp 133	. . . . . . 7 ⊢ (∪A ∈ ℘ x → ∪A ⊆ x)
31	28, 30	anim12i 268	. . . . . 6 ⊢ ((x ∈ A ∧ ∪A ∈ ℘ x) → (x ⊆ ∪A ∧ ∪A ⊆ x))
32		eqss 1516	. . . . . 6 ⊢ (x = ∪A ↔ (x ⊆ ∪A ∧ ∪A ⊆ x))
33	31, 32	sylibr 175	. . . . 5 ⊢ ((x ∈ A ∧ ∪A ∈ ℘ x) → x = ∪A)
34	33	exp 291	. . . 4 ⊢ (x ∈ A → (∪A ∈ ℘ x → x = ∪A))
35	34	r19.22i 1273	. . 3 ⊢ (∃x ∈ A ∪A ∈ ℘ x → ∃x ∈ A x = ∪A)
36	27, 35	syl 12	. 2 ⊢ (℘∪A = ∪x ∈ A ℘x → ∃x ∈ A x = ∪A)
37	18, 36	impbi 139	1 ⊢ (∃x ∈ A x = ∪A ↔ ℘∪A = ∪x ∈ A ℘x)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∃wrex 1202  Vcvv 1348   ⊆ wss 1487  ℘cpw 1798  ∪cuni 1919  ∪ciun 1994

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-in 1491  df-ss 1492  df-pw 1799  df-uni 1920  df-iun 1996

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