Theorem epttop 21614

Description: The excluded point topology. (Contributed by Mario Carneiro, 3-Sep-2015.)

Assertion

Ref	Expression
epttop	⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ (TopOn‘𝐴))

Distinct variable groups:   𝑥,𝐴   𝑥,𝑃

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem epttop

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab 4000	. . . . 5 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ↔ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)))
2		eleq2 2878	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ ∪ 𝑦))
3		eqeq1 2802	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 = 𝐴 ↔ ∪ 𝑦 = 𝐴))
4	2, 3	imbi12d 348	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑦 → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ ∪ 𝑦 → ∪ 𝑦 = 𝐴)))
5		simprl 770	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → 𝑦 ⊆ 𝒫 𝐴)
6		sspwuni 4985	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
7	5, 6	sylib 221	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → ∪ 𝑦 ⊆ 𝐴)
8		vuniex 7445	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ V
9	8	elpw 4501	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
10	7, 9	sylibr 237	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → ∪ 𝑦 ∈ 𝒫 𝐴)
11		eluni2 4804	. . . . . . . . . 10 ⊢ (𝑃 ∈ ∪ 𝑦 ↔ ∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥)
12		r19.29 3216	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ ∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥) → ∃𝑥 ∈ 𝑦 ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥))
13		simpr 488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)) → (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))
14	13	impr 458	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥)) → 𝑥 = 𝐴)
15		elssuni 4830	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝑦)
16	15	adantr 484	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥)) → 𝑥 ⊆ ∪ 𝑦)
17	14, 16	eqsstrrd 3954	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥)) → 𝐴 ⊆ ∪ 𝑦)
18	17	rexlimiva 3240	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝑦 ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥) → 𝐴 ⊆ ∪ 𝑦)
19	12, 18	syl 17	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ ∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥) → 𝐴 ⊆ ∪ 𝑦)
20	19	ex 416	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴) → (∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥 → 𝐴 ⊆ ∪ 𝑦))
21	20	ad2antll 728	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → (∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥 → 𝐴 ⊆ ∪ 𝑦))
22	11, 21	syl5bi 245	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → (𝑃 ∈ ∪ 𝑦 → 𝐴 ⊆ ∪ 𝑦))
23	22, 7	jctild 529	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → (𝑃 ∈ ∪ 𝑦 → (∪ 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦)))
24		eqss 3930	. . . . . . . 8 ⊢ (∪ 𝑦 = 𝐴 ↔ (∪ 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))
25	23, 24	syl6ibr 255	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → (𝑃 ∈ ∪ 𝑦 → ∪ 𝑦 = 𝐴))
26	4, 10, 25	elrabd 3630	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})
27	26	ex 416	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ((𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)) → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}))
28	1, 27	syl5bi 245	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}))
29	28	alrimiv 1928	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}))
30		inss1 4155	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝑧) ⊆ 𝑦
31		simprll 778	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → 𝑦 ∈ 𝒫 𝐴)
32	31	elpwid 4508	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → 𝑦 ⊆ 𝐴)
33	30, 32	sstrid 3926	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → (𝑦 ∩ 𝑧) ⊆ 𝐴)
34		vex 3444	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
35	34	inex1 5185	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝑧) ∈ V
36	35	elpw 4501	. . . . . . . 8 ⊢ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ↔ (𝑦 ∩ 𝑧) ⊆ 𝐴)
37	33, 36	sylibr 237	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → (𝑦 ∩ 𝑧) ∈ 𝒫 𝐴)
38		elin 3897	. . . . . . . 8 ⊢ (𝑃 ∈ (𝑦 ∩ 𝑧) ↔ (𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧))
39		simprlr 779	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → (𝑃 ∈ 𝑦 → 𝑦 = 𝐴))
40		simprrr 781	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → (𝑃 ∈ 𝑧 → 𝑧 = 𝐴))
41	39, 40	anim12d 611	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → ((𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧) → (𝑦 = 𝐴 ∧ 𝑧 = 𝐴)))
42		ineq12 4134	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → (𝑦 ∩ 𝑧) = (𝐴 ∩ 𝐴))
43		inidm 4145	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐴) = 𝐴
44	42, 43	eqtrdi 2849	. . . . . . . . 9 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → (𝑦 ∩ 𝑧) = 𝐴)
45	41, 44	syl6 35	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → ((𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧) → (𝑦 ∩ 𝑧) = 𝐴))
46	38, 45	syl5bi 245	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → (𝑃 ∈ (𝑦 ∩ 𝑧) → (𝑦 ∩ 𝑧) = 𝐴))
47	37, 46	jca 515	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦 ∩ 𝑧) → (𝑦 ∩ 𝑧) = 𝐴)))
48	47	ex 416	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴))) → ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦 ∩ 𝑧) → (𝑦 ∩ 𝑧) = 𝐴))))
49		eleq2 2878	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
50		eqeq1 2802	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
51	49, 50	imbi12d 348	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)))
52	51	elrab 3628	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)))
53		eleq2 2878	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑧))
54		eqeq1 2802	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐴 ↔ 𝑧 = 𝐴))
55	53, 54	imbi12d 348	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))
56	55	elrab 3628	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))
57	52, 56	anbi12i 629	. . . . 5 ⊢ ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}) ↔ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴))))
58		eleq2 2878	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑦 ∩ 𝑧)))
59		eqeq1 2802	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑥 = 𝐴 ↔ (𝑦 ∩ 𝑧) = 𝐴))
60	58, 59	imbi12d 348	. . . . . 6 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ (𝑦 ∩ 𝑧) → (𝑦 ∩ 𝑧) = 𝐴)))
61	60	elrab 3628	. . . . 5 ⊢ ((𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ↔ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦 ∩ 𝑧) → (𝑦 ∩ 𝑧) = 𝐴)))
62	48, 57, 61	3imtr4g 299	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ((𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∧ 𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}) → (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}))
63	62	ralrimivv 3155	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})
64		pwexg 5244	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
65	64	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝒫 𝐴 ∈ V)
66		rabexg 5198	. . . . 5 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ V)
67	65, 66	syl 17	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ V)
68		istopg 21500	. . . 4 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ V → ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})))
69	67, 68	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ Top ↔ (∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}) ∧ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}∀𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} (𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})))
70	29, 63, 69	mpbir2and 712	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ Top)
71		eleq2 2878	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝐴))
72		eqeq1 2802	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
73	71, 72	imbi12d 348	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ 𝐴 → 𝐴 = 𝐴)))
74		pwidg 4519	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
75	74	adantr 484	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ 𝒫 𝐴)
76		eqidd 2799	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 = 𝐴)
77	76	a1d 25	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑃 ∈ 𝐴 → 𝐴 = 𝐴))
78	73, 75, 77	elrabd 3630	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})
79		elssuni 4830	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} → 𝐴 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})
80	78, 79	syl 17	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})
81		ssrab2 4007	. . . . 5 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ⊆ 𝒫 𝐴
82		sspwuni 4985	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ⊆ 𝐴)
83	81, 82	mpbi 233	. . . 4 ⊢ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ⊆ 𝐴
84	83	a1i 11	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ⊆ 𝐴)
85	80, 84	eqssd 3932	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 = ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})
86		istopon 21517	. 2 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ (TopOn‘𝐴) ↔ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ Top ∧ 𝐴 = ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}))
87	70, 85, 86	sylanbrc 586	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ (TopOn‘𝐴))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399  ∀wal 1536   = wceq 1538   ∈ wcel 2111  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∩ cin 3880   ⊆ wss 3881  𝒫 cpw 4497  ∪ cuni 4800  ‘cfv 6324  Topctop 21498  TopOnctopon 21515

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-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-iota 6283  df-fun 6326  df-fv 6332  df-top 21499  df-topon 21516

This theorem is referenced by:  dfac14lem  22222  dfac14  22223