Theorem epttop 13170

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 3231	. . . . 5 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ↔ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)))
2		simprl 529	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → 𝑦 ⊆ 𝒫 𝐴)
3		sspwuni 3966	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
4	2, 3	sylib 122	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → ∪ 𝑦 ⊆ 𝐴)
5		vuniex 4432	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ V
6	5	elpw 3578	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
7	4, 6	sylibr 134	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → ∪ 𝑦 ∈ 𝒫 𝐴)
8		eluni2 3809	. . . . . . . . . 10 ⊢ (𝑃 ∈ ∪ 𝑦 ↔ ∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥)
9		r19.29 2612	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ ∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥) → ∃𝑥 ∈ 𝑦 ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥))
10		simpr 110	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)) → (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))
11	10	impr 379	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥)) → 𝑥 = 𝐴)
12		elssuni 3833	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝑦)
13	12	adantr 276	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥)) → 𝑥 ⊆ ∪ 𝑦)
14	11, 13	eqsstrrd 3190	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥)) → 𝐴 ⊆ ∪ 𝑦)
15	14	rexlimiva 2587	. . . . . . . . . . . . 13 ⊢ (∃𝑥 ∈ 𝑦 ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥) → 𝐴 ⊆ ∪ 𝑦)
16	9, 15	syl 14	. . . . . . . . . . . 12 ⊢ ((∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ ∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥) → 𝐴 ⊆ ∪ 𝑦)
17	16	ex 115	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴) → (∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥 → 𝐴 ⊆ ∪ 𝑦))
18	17	ad2antll 491	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → (∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥 → 𝐴 ⊆ ∪ 𝑦))
19	8, 18	biimtrid 152	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → (𝑃 ∈ ∪ 𝑦 → 𝐴 ⊆ ∪ 𝑦))
20	19, 4	jctild 316	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → (𝑃 ∈ ∪ 𝑦 → (∪ 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦)))
21		eqss 3168	. . . . . . . 8 ⊢ (∪ 𝑦 = 𝐴 ↔ (∪ 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))
22	20, 21	syl6ibr 162	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → (𝑃 ∈ ∪ 𝑦 → ∪ 𝑦 = 𝐴))
23		eleq2 2239	. . . . . . . . 9 ⊢ (𝑥 = ∪ 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ ∪ 𝑦))
24		eqeq1 2182	. . . . . . . . 9 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 = 𝐴 ↔ ∪ 𝑦 = 𝐴))
25	23, 24	imbi12d 234	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ ∪ 𝑦 → ∪ 𝑦 = 𝐴)))
26	25	elrab 2891	. . . . . . 7 ⊢ (∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ↔ (∪ 𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ ∪ 𝑦 → ∪ 𝑦 = 𝐴)))
27	7, 22, 26	sylanbrc 417	. . . . . 6 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})
28	27	ex 115	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ((𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)) → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}))
29	1, 28	biimtrid 152	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}))
30	29	alrimiv 1872	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}))
31		inss1 3353	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝑧) ⊆ 𝑦
32		simprll 537	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → 𝑦 ∈ 𝒫 𝐴)
33	32	elpwid 3583	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → 𝑦 ⊆ 𝐴)
34	31, 33	sstrid 3164	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → (𝑦 ∩ 𝑧) ⊆ 𝐴)
35		vex 2738	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
36	35	inex1 4132	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝑧) ∈ V
37	36	elpw 3578	. . . . . . . 8 ⊢ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ↔ (𝑦 ∩ 𝑧) ⊆ 𝐴)
38	34, 37	sylibr 134	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → (𝑦 ∩ 𝑧) ∈ 𝒫 𝐴)
39		elin 3316	. . . . . . . 8 ⊢ (𝑃 ∈ (𝑦 ∩ 𝑧) ↔ (𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧))
40		simprlr 538	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → (𝑃 ∈ 𝑦 → 𝑦 = 𝐴))
41		simprrr 540	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → (𝑃 ∈ 𝑧 → 𝑧 = 𝐴))
42	40, 41	anim12d 335	. . . . . . . . 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)))
43		ineq12 3329	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → (𝑦 ∩ 𝑧) = (𝐴 ∩ 𝐴))
44		inidm 3342	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐴) = 𝐴
45	43, 44	eqtrdi 2224	. . . . . . . . 9 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → (𝑦 ∩ 𝑧) = 𝐴)
46	42, 45	syl6 33	. . . . . . . 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))
47	39, 46	biimtrid 152	. . . . . . 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))
48	38, 47	jca 306	. . . . . 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)))
49	48	ex 115	. . . . 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))))
50		eleq2 2239	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
51		eqeq1 2182	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
52	50, 51	imbi12d 234	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)))
53	52	elrab 2891	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)))
54		eleq2 2239	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑧))
55		eqeq1 2182	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐴 ↔ 𝑧 = 𝐴))
56	54, 55	imbi12d 234	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))
57	56	elrab 2891	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))
58	53, 57	anbi12i 460	. . . . 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))))
59		eleq2 2239	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑦 ∩ 𝑧)))
60		eqeq1 2182	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑥 = 𝐴 ↔ (𝑦 ∩ 𝑧) = 𝐴))
61	59, 60	imbi12d 234	. . . . . 6 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ (𝑦 ∩ 𝑧) → (𝑦 ∩ 𝑧) = 𝐴)))
62	61	elrab 2891	. . . . 5 ⊢ ((𝑦 ∩ 𝑧) ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ↔ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ∧ (𝑃 ∈ (𝑦 ∩ 𝑧) → (𝑦 ∩ 𝑧) = 𝐴)))
63	49, 58, 62	3imtr4g 205	. . . 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)}))
64	63	ralrimivv 2556	. . 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)})
65		pwexg 4175	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
66	65	adantr 276	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝒫 𝐴 ∈ V)
67		rabexg 4141	. . . . 5 ⊢ (𝒫 𝐴 ∈ V → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ V)
68	66, 67	syl 14	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ V)
69		istopg 13077	. . . 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)})))
70	68, 69	syl 14	. . 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)})))
71	30, 64, 70	mpbir2and 944	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ Top)
72		pwidg 3586	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
73	72	adantr 276	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ 𝒫 𝐴)
74		eqidd 2176	. . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 = 𝐴)
75	74	a1d 22	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑃 ∈ 𝐴 → 𝐴 = 𝐴))
76		eleq2 2239	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝐴))
77		eqeq1 2182	. . . . . . 7 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
78	76, 77	imbi12d 234	. . . . . 6 ⊢ (𝑥 = 𝐴 → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ 𝐴 → 𝐴 = 𝐴)))
79	78	elrab 2891	. . . . 5 ⊢ (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ↔ (𝐴 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝐴 → 𝐴 = 𝐴)))
80	73, 75, 79	sylanbrc 417	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})
81		elssuni 3833	. . . 4 ⊢ (𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} → 𝐴 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})
82	80, 81	syl 14	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ⊆ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})
83		ssrab2 3238	. . . . 5 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ⊆ 𝒫 𝐴
84		sspwuni 3966	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ⊆ 𝐴)
85	83, 84	mpbi 145	. . . 4 ⊢ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ⊆ 𝐴
86	85	a1i 9	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ⊆ 𝐴)
87	82, 86	eqssd 3170	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 = ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})
88		istopon 13091	. 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)}))
89	71, 87, 88	sylanbrc 417	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ (TopOn‘𝐴))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1351   = wceq 1353   ∈ wcel 2146  ∀wral 2453  ∃wrex 2454  {crab 2457  Vcvv 2735   ∩ cin 3126   ⊆ wss 3127  𝒫 cpw 3572  ∪ cuni 3805  ‘cfv 5208  Topctop 13075  TopOnctopon 13088

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-sbc 2961  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-mpt 4061  df-id 4287  df-xp 4626  df-rel 4627  df-cnv 4628  df-co 4629  df-dm 4630  df-iota 5170  df-fun 5210  df-fv 5216  df-top 13076  df-topon 13089

This theorem is referenced by: (None)