Theorem isreg2 23315

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 22852	. . . . . . 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 1138	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → ¬ 𝑥 ∈ 𝑐)
9		eqid 2735	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	regsep2 23314	. . . . 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 1374	. . . 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 1121	. . 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 3187	. 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 22851	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
15	14	adantr 480	. . 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 480	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → 𝑋 = ∪ 𝐽)
17	16	difeq1d 4100	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (𝑋 ∖ 𝑦) = (∪ 𝐽 ∖ 𝑦))
18	9	opncld 22971	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
19	14, 18	sylan 580	. . . . . . . 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 3936	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑋 ∖ 𝑦) ↔ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ 𝑦))
24	23	baibr 536	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑋 ∖ 𝑦)))
25	24	con1bid 355	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ (𝑋 ∖ 𝑦) ↔ 𝑥 ∈ 𝑦))
26	22, 25	sylan9bb 509	. . . . . . . . . 10 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → (¬ 𝑥 ∈ 𝑐 ↔ 𝑥 ∈ 𝑦))
27		simpl 482	. . . . . . . . . . . . 13 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → 𝑐 = (𝑋 ∖ 𝑦))
28	27	sseq1d 3990	. . . . . . . . . . . 12 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → (𝑐 ⊆ 𝑜 ↔ (𝑋 ∖ 𝑦) ⊆ 𝑜))
29	28	3anbi1d 1442	. . . . . . . . . . 11 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → ((𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅) ↔ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)))
30	29	2rexbidv 3206	. . . . . . . . . 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 344	. . . . . . . . 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 3161	. . . . . . . 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 3597	. . . . . . 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 3086	. . . . . . 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 22865	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → 𝑦 ⊆ 𝑋)
37	36	sselda 3958	. . . . . . . . 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 730	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑋 = ∪ 𝐽)
40	39	difeq1d 4100	. . . . . . . . . . . . . . . . 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 730	. . . . . . . . . . . . . . . . . 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 22971	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑜) ∈ (Clsd‘𝐽))
44	41, 42, 43	syl2anc 584	. . . . . . . . . . . . . . . . 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 4184	. . . . . . . . . . . . . . . . . 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 2782	. . . . . . . . . . . . . . . . 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 22865	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑝 ∈ 𝐽) → 𝑝 ⊆ 𝑋)
52	49, 50, 51	syl2anc 584	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑝 ⊆ 𝑋)
53		reldisj 4428	. . . . . . . . . . . . . . . . . 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 232	. . . . . . . . . . . . . . . 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 23010	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽) ∧ 𝑝 ⊆ (𝑋 ∖ 𝑜)) → ((cls‘𝐽)‘𝑝) ⊆ (𝑋 ∖ 𝑜))
57	45, 55, 56	syl2anc 584	. . . . . . . . . . . . . . 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 4464	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑦)
60	58, 59	sylib 218	. . . . . . . . . . . . . . 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 3969	. . . . . . . . . . . . . 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 511	. . . . . . . . . . . . 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 456	. . . . . . . . . . . 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 467	. . . . . . . . . . 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 3153	. . . . . . . . . 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 3141	. . . . . . . . 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 3152	. . . . . . 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 242	. . . . . 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 3140	. . . 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 406	. . 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 23270	. . 3 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝐽 ∀𝑥 ∈ 𝑦 ∃𝑝 ∈ 𝐽 (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
74	15, 72, 73	sylanbrc 583	. 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) = ∅))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2108  ∀wral 3051  ∃wrex 3060   ∖ cdif 3923   ∩ cin 3925   ⊆ wss 3926  ∅c0 4308  ∪ cuni 4883  ‘cfv 6531  Topctop 22831  TopOnctopon 22848  Clsdccld 22954  clsccl 22956  Regcreg 23247

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-rep 5249  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-reu 3360  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-int 4923  df-iun 4969  df-iin 4970  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-top 22832  df-topon 22849  df-cld 22957  df-cls 22959  df-reg 23254

This theorem is referenced by: (None)