Theorem epttop 21324

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 3941	. . . . 5 ⊢ (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ↔ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)))
2		eleq2 2854	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ ∪ 𝑦))
3		eqeq1 2782	. . . . . . . 8 ⊢ (𝑥 = ∪ 𝑦 → (𝑥 = 𝐴 ↔ ∪ 𝑦 = 𝐴))
4	2, 3	imbi12d 337	. . . . . . 7 ⊢ (𝑥 = ∪ 𝑦 → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ ∪ 𝑦 → ∪ 𝑦 = 𝐴)))
5		simprl 758	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → 𝑦 ⊆ 𝒫 𝐴)
6		sspwuni 4889	. . . . . . . . 9 ⊢ (𝑦 ⊆ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
7	5, 6	sylib 210	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → ∪ 𝑦 ⊆ 𝐴)
8		vuniex 7286	. . . . . . . . 9 ⊢ ∪ 𝑦 ∈ V
9	8	elpw 4429	. . . . . . . 8 ⊢ (∪ 𝑦 ∈ 𝒫 𝐴 ↔ ∪ 𝑦 ⊆ 𝐴)
10	7, 9	sylibr 226	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → ∪ 𝑦 ∈ 𝒫 𝐴)
11		eluni2 4717	. . . . . . . . . 10 ⊢ (𝑃 ∈ ∪ 𝑦 ↔ ∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥)
12		r19.29 3200	. . . . . . . . . . . . 13 ⊢ ((∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ ∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥) → ∃𝑥 ∈ 𝑦 ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥))
13		simpr 477	. . . . . . . . . . . . . . . 16 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)) → (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))
14	13	impr 447	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥)) → 𝑥 = 𝐴)
15		elssuni 4742	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 ∈ 𝑦 → 𝑥 ⊆ ∪ 𝑦)
16	15	adantr 473	. . . . . . . . . . . . . . 15 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥)) → 𝑥 ⊆ ∪ 𝑦)
17	14, 16	eqsstr3d 3898	. . . . . . . . . . . . . 14 ⊢ ((𝑥 ∈ 𝑦 ∧ ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ∧ 𝑃 ∈ 𝑥)) → 𝐴 ⊆ ∪ 𝑦)
18	17	rexlimiva 3226	. . . . . . . . . . . . 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 405	. . . . . . . . . . 11 ⊢ (∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴) → (∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥 → 𝐴 ⊆ ∪ 𝑦))
21	20	ad2antll 716	. . . . . . . . . 10 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → (∃𝑥 ∈ 𝑦 𝑃 ∈ 𝑥 → 𝐴 ⊆ ∪ 𝑦))
22	11, 21	syl5bi 234	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → (𝑃 ∈ ∪ 𝑦 → 𝐴 ⊆ ∪ 𝑦))
23	22, 7	jctild 518	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → (𝑃 ∈ ∪ 𝑦 → (∪ 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦)))
24		eqss 3875	. . . . . . . 8 ⊢ (∪ 𝑦 = 𝐴 ↔ (∪ 𝑦 ⊆ 𝐴 ∧ 𝐴 ⊆ ∪ 𝑦))
25	23, 24	syl6ibr 244	. . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ (𝑦 ⊆ 𝒫 𝐴 ∧ ∀𝑥 ∈ 𝑦 (𝑃 ∈ 𝑥 → 𝑥 = 𝐴))) → (𝑃 ∈ ∪ 𝑦 → ∪ 𝑦 = 𝐴))
26	4, 10, 25	elrabd 3598	. . . . . 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 405	. . . . 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 234	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → (𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}))
29	28	alrimiv 1886	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → ∀𝑦(𝑦 ⊆ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} → ∪ 𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)}))
30		inss1 4094	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝑧) ⊆ 𝑦
31		simprll 766	. . . . . . . . . 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 4435	. . . . . . . . 9 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → 𝑦 ⊆ 𝐴)
33	30, 32	syl5ss 3871	. . . . . . . 8 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) ∧ ((𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)) ∧ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))) → (𝑦 ∩ 𝑧) ⊆ 𝐴)
34		vex 3418	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
35	34	inex1 5079	. . . . . . . . 9 ⊢ (𝑦 ∩ 𝑧) ∈ V
36	35	elpw 4429	. . . . . . . 8 ⊢ ((𝑦 ∩ 𝑧) ∈ 𝒫 𝐴 ↔ (𝑦 ∩ 𝑧) ⊆ 𝐴)
37	33, 36	sylibr 226	. . . . . . 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 4059	. . . . . . . 8 ⊢ (𝑃 ∈ (𝑦 ∩ 𝑧) ↔ (𝑃 ∈ 𝑦 ∧ 𝑃 ∈ 𝑧))
39		simprlr 767	. . . . . . . . . 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 769	. . . . . . . . . 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 599	. . . . . . . . 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 4073	. . . . . . . . . 10 ⊢ ((𝑦 = 𝐴 ∧ 𝑧 = 𝐴) → (𝑦 ∩ 𝑧) = (𝐴 ∩ 𝐴))
43		inidm 4084	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐴) = 𝐴
44	42, 43	syl6eq 2830	. . . . . . . . 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 234	. . . . . . 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 504	. . . . . 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 405	. . . . 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 2854	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑦))
50		eqeq1 2782	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (𝑥 = 𝐴 ↔ 𝑦 = 𝐴))
51	49, 50	imbi12d 337	. . . . . . 7 ⊢ (𝑥 = 𝑦 → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)))
52	51	elrab 3595	. . . . . 6 ⊢ (𝑦 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ↔ (𝑦 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑦 → 𝑦 = 𝐴)))
53		eleq2 2854	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝑧))
54		eqeq1 2782	. . . . . . . 8 ⊢ (𝑥 = 𝑧 → (𝑥 = 𝐴 ↔ 𝑧 = 𝐴))
55	53, 54	imbi12d 337	. . . . . . 7 ⊢ (𝑥 = 𝑧 → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))
56	55	elrab 3595	. . . . . 6 ⊢ (𝑧 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ↔ (𝑧 ∈ 𝒫 𝐴 ∧ (𝑃 ∈ 𝑧 → 𝑧 = 𝐴)))
57	52, 56	anbi12i 617	. . . . 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 2854	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ (𝑦 ∩ 𝑧)))
59		eqeq1 2782	. . . . . . 7 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → (𝑥 = 𝐴 ↔ (𝑦 ∩ 𝑧) = 𝐴))
60	58, 59	imbi12d 337	. . . . . 6 ⊢ (𝑥 = (𝑦 ∩ 𝑧) → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ (𝑦 ∩ 𝑧) → (𝑦 ∩ 𝑧) = 𝐴)))
61	60	elrab 3595	. . . . 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 288	. . . 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 3140	. . 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 5133	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝒫 𝐴 ∈ V)
65	64	adantr 473	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝒫 𝐴 ∈ V)
66		rabexg 5091	. . . . 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 21210	. . . 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 700	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ Top)
71		eleq2 2854	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑃 ∈ 𝑥 ↔ 𝑃 ∈ 𝐴))
72		eqeq1 2782	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥 = 𝐴 ↔ 𝐴 = 𝐴))
73	71, 72	imbi12d 337	. . . . 5 ⊢ (𝑥 = 𝐴 → ((𝑃 ∈ 𝑥 → 𝑥 = 𝐴) ↔ (𝑃 ∈ 𝐴 → 𝐴 = 𝐴)))
74		pwidg 4438	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ 𝒫 𝐴)
75	74	adantr 473	. . . . 5 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ 𝒫 𝐴)
76		eqidd 2779	. . . . . 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 3598	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 ∈ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})
79		elssuni 4742	. . . 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 3948	. . . . 5 ⊢ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ⊆ 𝒫 𝐴
82		sspwuni 4889	. . . . 5 ⊢ ({𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ⊆ 𝒫 𝐴 ↔ ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ⊆ 𝐴)
83	81, 82	mpbi 222	. . . 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 3877	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → 𝐴 = ∪ {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)})
86		istopon 21227	. 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 575	1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑃 ∈ 𝐴) → {𝑥 ∈ 𝒫 𝐴 ∣ (𝑃 ∈ 𝑥 → 𝑥 = 𝐴)} ∈ (TopOn‘𝐴))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387  ∀wal 1505   = wceq 1507   ∈ wcel 2050  ∀wral 3088  ∃wrex 3089  {crab 3092  Vcvv 3415   ∩ cin 3830   ⊆ wss 3831  𝒫 cpw 4423  ∪ cuni 4713  ‘cfv 6190  Topctop 21208  TopOnctopon 21225

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3684  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-op 4449  df-uni 4714  df-br 4931  df-opab 4993  df-mpt 5010  df-id 5313  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-iota 6154  df-fun 6192  df-fv 6198  df-top 21209  df-topon 21226

This theorem is referenced by:  dfac14lem  21932  dfac14  21933