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Theorem isreg2 22873

Description: A topological space is regular if any closed set is separated from any point not in it by neighborhoods. (Contributed by Jeff Hankins, 1-Feb-2010.) (Revised by Mario Carneiro, 25-Aug-2015.)

**Assertion**
Ref	Expression
isreg2	⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Reg ↔ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))))

Distinct variable groups: 𝑜,𝑐,𝑝,𝑥,𝐽 𝑋,𝑐,𝑜,𝑝,𝑥

Proof of Theorem isreg2 Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simp1r 1199	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝐽 ∈ Reg)
2		simp2l 1200	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝑐 ∈ (Clsd‘𝐽))
3		simp2r 1201	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝑥 ∈ 𝑋)
4		simp1l 1198	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝐽 ∈ (TopOn‘𝑋))
5		toponuni 22408	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
6	4, 5	syl 17	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝑋 = ∪ 𝐽)
7	3, 6	eleqtrd 2836	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝑥 ∈ ∪ 𝐽)
8		simp3 1139	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → ¬ 𝑥 ∈ 𝑐)
9		eqid 2733	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	regsep2 22872	. . . . 5 ⊢ ((𝐽 ∈ Reg ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ ∪ 𝐽 ∧ ¬ 𝑥 ∈ 𝑐)) → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))
11	1, 2, 7, 8, 10	syl13anc 1373	. . . 4 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))
12	11	3expia 1122	. . 3 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋)) → (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)))
13	12	ralrimivva 3201	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)))
14		topontop 22407	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
15	14	adantr 482	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝐽 ∈ Top)
16	5	adantr 482	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → 𝑋 = ∪ 𝐽)
17	16	difeq1d 4121	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (𝑋 ∖ 𝑦) = (∪ 𝐽 ∖ 𝑦))
18	9	opncld 22529	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
19	14, 18	sylan 581	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
20	17, 19	eqeltrd 2834	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (𝑋 ∖ 𝑦) ∈ (Clsd‘𝐽))
21		eleq2 2823	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑋 ∖ 𝑦) → (𝑥 ∈ 𝑐 ↔ 𝑥 ∈ (𝑋 ∖ 𝑦)))
22	21	notbid 318	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑋 ∖ 𝑦) → (¬ 𝑥 ∈ 𝑐 ↔ ¬ 𝑥 ∈ (𝑋 ∖ 𝑦)))
23		eldif 3958	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑋 ∖ 𝑦) ↔ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ 𝑦))
24	23	baibr 538	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑋 ∖ 𝑦)))
25	24	con1bid 356	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ (𝑋 ∖ 𝑦) ↔ 𝑥 ∈ 𝑦))
26	22, 25	sylan9bb 511	. . . . . . . . . 10 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → (¬ 𝑥 ∈ 𝑐 ↔ 𝑥 ∈ 𝑦))
27		simpl 484	. . . . . . . . . . . . 13 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → 𝑐 = (𝑋 ∖ 𝑦))
28	27	sseq1d 4013	. . . . . . . . . . . 12 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → (𝑐 ⊆ 𝑜 ↔ (𝑋 ∖ 𝑦) ⊆ 𝑜))
29	28	3anbi1d 1441	. . . . . . . . . . 11 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → ((𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅) ↔ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)))
30	29	2rexbidv 3220	. . . . . . . . . 10 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → (∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅) ↔ ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)))
31	26, 30	imbi12d 345	. . . . . . . . 9 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → ((¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)) ↔ (𝑥 ∈ 𝑦 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))))
32	31	ralbidva 3176	. . . . . . . 8 ⊢ (𝑐 = (𝑋 ∖ 𝑦) → (∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)) ↔ ∀𝑥 ∈ 𝑋 (𝑥 ∈ 𝑦 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))))
33	32	rspcv 3609	. . . . . . 7 ⊢ ((𝑋 ∖ 𝑦) ∈ (Clsd‘𝐽) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)) → ∀𝑥 ∈ 𝑋 (𝑥 ∈ 𝑦 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))))
34	20, 33	syl 17	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)) → ∀𝑥 ∈ 𝑋 (𝑥 ∈ 𝑦 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))))
35		ralcom3 3098	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝑋 (𝑥 ∈ 𝑦 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)) ↔ ∀𝑥 ∈ 𝑦 (𝑥 ∈ 𝑋 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)))
36		toponss 22421	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → 𝑦 ⊆ 𝑋)
37	36	sselda 3982	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝑋)
38		simprr2 1223	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑥 ∈ 𝑝)
39	5	ad3antrrr 729	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑋 = ∪ 𝐽)
40	39	difeq1d 4121	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → (𝑋 ∖ 𝑜) = (∪ 𝐽 ∖ 𝑜))
41	14	ad3antrrr 729	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝐽 ∈ Top)
42		simprll 778	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑜 ∈ 𝐽)
43	9	opncld 22529	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑜) ∈ (Clsd‘𝐽))
44	41, 42, 43	syl2anc 585	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → (∪ 𝐽 ∖ 𝑜) ∈ (Clsd‘𝐽))
45	40, 44	eqeltrd 2834	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → (𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽))
46		incom 4201	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∩ 𝑜) = (𝑜 ∩ 𝑝)
47		simprr3 1224	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → (𝑜 ∩ 𝑝) = ∅)
48	46, 47	eqtrid 2785	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → (𝑝 ∩ 𝑜) = ∅)
49		simplll 774	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝐽 ∈ (TopOn‘𝑋))
50		simprlr 779	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑝 ∈ 𝐽)
51		toponss 22421	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑝 ∈ 𝐽) → 𝑝 ⊆ 𝑋)
52	49, 50, 51	syl2anc 585	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑝 ⊆ 𝑋)
53		reldisj 4451	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ⊆ 𝑋 → ((𝑝 ∩ 𝑜) = ∅ ↔ 𝑝 ⊆ (𝑋 ∖ 𝑜)))
54	52, 53	syl 17	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → ((𝑝 ∩ 𝑜) = ∅ ↔ 𝑝 ⊆ (𝑋 ∖ 𝑜)))
55	48, 54	mpbid 231	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑝 ⊆ (𝑋 ∖ 𝑜))
56	9	clsss2 22568	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽) ∧ 𝑝 ⊆ (𝑋 ∖ 𝑜)) → ((cls‘𝐽)‘𝑝) ⊆ (𝑋 ∖ 𝑜))
57	45, 55, 56	syl2anc 585	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → ((cls‘𝐽)‘𝑝) ⊆ (𝑋 ∖ 𝑜))
58		simprr1 1222	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → (𝑋 ∖ 𝑦) ⊆ 𝑜)
59		difcom 4488	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑦)
60	58, 59	sylib 217	. . . . . . . . . . . . . . 15 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → (𝑋 ∖ 𝑜) ⊆ 𝑦)
61	57, 60	sstrd 3992	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → ((cls‘𝐽)‘𝑝) ⊆ 𝑦)
62	38, 61	jca 513	. . . . . . . . . . . . 13 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦))
63	62	expr 458	. . . . . . . . . . . 12 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ (𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽)) → (((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅) → (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
64	63	anassrs 469	. . . . . . . . . . 11 ⊢ (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ 𝑜 ∈ 𝐽) ∧ 𝑝 ∈ 𝐽) → (((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅) → (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
65	64	reximdva 3169	. . . . . . . . . 10 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ 𝑜 ∈ 𝐽) → (∃𝑝 ∈ 𝐽 ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅) → ∃𝑝 ∈ 𝐽 (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
66	65	rexlimdva 3156	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) → (∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅) → ∃𝑝 ∈ 𝐽 (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
67	37, 66	embantd 59	. . . . . . . 8 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) → ((𝑥 ∈ 𝑋 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)) → ∃𝑝 ∈ 𝐽 (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
68	67	ralimdva 3168	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (∀𝑥 ∈ 𝑦 (𝑥 ∈ 𝑋 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)) → ∀𝑥 ∈ 𝑦 ∃𝑝 ∈ 𝐽 (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
69	35, 68	biimtrid 241	. . . . . 6 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (∀𝑥 ∈ 𝑋 (𝑥 ∈ 𝑦 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)) → ∀𝑥 ∈ 𝑦 ∃𝑝 ∈ 𝐽 (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
70	34, 69	syld 47	. . . . 5 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)) → ∀𝑥 ∈ 𝑦 ∃𝑝 ∈ 𝐽 (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
71	70	ralrimdva 3155	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)) → ∀𝑦 ∈ 𝐽 ∀𝑥 ∈ 𝑦 ∃𝑝 ∈ 𝐽 (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
72	71	imp 408	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → ∀𝑦 ∈ 𝐽 ∀𝑥 ∈ 𝑦 ∃𝑝 ∈ 𝐽 (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦))
73		isreg 22828	. . 3 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝐽 ∀𝑥 ∈ 𝑦 ∃𝑝 ∈ 𝐽 (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
74	15, 72, 73	sylanbrc 584	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝐽 ∈ Reg)
75	13, 74	impbida 800	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Reg ↔ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ∃wrex 3071 ∖ cdif 3945 ∩ cin 3947 ⊆ wss 3948 ∅c0 4322 ∪ cuni 4908 ‘cfv 6541 Topctop 22387 TopOnctopon 22404 Clsdccld 22512 clsccl 22514 Regcreg 22805

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5285 ax-sep 5299 ax-nul 5306 ax-pow 5363 ax-pr 5427 ax-un 7722

This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3778 df-csb 3894 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-op 4635 df-uni 4909 df-int 4951 df-iun 4999 df-iin 5000 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5574 df-xp 5682 df-rel 5683 df-cnv 5684 df-co 5685 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-iota 6493 df-fun 6543 df-fn 6544 df-f 6545 df-f1 6546 df-fo 6547 df-f1o 6548 df-fv 6549 df-top 22388 df-topon 22405 df-cld 22515 df-cls 22517 df-reg 22812

This theorem is referenced by: (None)

W3C validator