Theorem isreg2 21181

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 1086	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝐽 ∈ Reg)
2		simp2l 1087	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝑐 ∈ (Clsd‘𝐽))
3		simp2r 1088	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝑥 ∈ 𝑋)
4		simp1l 1085	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝐽 ∈ (TopOn‘𝑋))
5		toponuni 20719	. . . . . . 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 2703	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝑥 ∈ ∪ 𝐽)
8		simp3 1063	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → ¬ 𝑥 ∈ 𝑐)
9		eqid 2622	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	regsep2 21180	. . . . 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 1328	. . . 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 1267	. . 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 2971	. 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 20718	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
15	14	adantr 481	. . 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 481	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → 𝑋 = ∪ 𝐽)
17	16	difeq1d 3727	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (𝑋 ∖ 𝑦) = (∪ 𝐽 ∖ 𝑦))
18	9	opncld 20837	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
19	14, 18	sylan 488	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
20	17, 19	eqeltrd 2701	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (𝑋 ∖ 𝑦) ∈ (Clsd‘𝐽))
21		eleq2 2690	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑋 ∖ 𝑦) → (𝑥 ∈ 𝑐 ↔ 𝑥 ∈ (𝑋 ∖ 𝑦)))
22	21	notbid 308	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑋 ∖ 𝑦) → (¬ 𝑥 ∈ 𝑐 ↔ ¬ 𝑥 ∈ (𝑋 ∖ 𝑦)))
23		eldif 3584	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑋 ∖ 𝑦) ↔ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ 𝑦))
24	23	baibr 945	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑋 ∖ 𝑦)))
25	24	con1bid 345	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ (𝑋 ∖ 𝑦) ↔ 𝑥 ∈ 𝑦))
26	22, 25	sylan9bb 736	. . . . . . . . . 10 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → (¬ 𝑥 ∈ 𝑐 ↔ 𝑥 ∈ 𝑦))
27		simpl 473	. . . . . . . . . . . . 13 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → 𝑐 = (𝑋 ∖ 𝑦))
28	27	sseq1d 3632	. . . . . . . . . . . 12 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → (𝑐 ⊆ 𝑜 ↔ (𝑋 ∖ 𝑦) ⊆ 𝑜))
29	28	3anbi1d 1403	. . . . . . . . . . 11 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → ((𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅) ↔ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)))
30	29	2rexbidv 3057	. . . . . . . . . 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 334	. . . . . . . . 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 2985	. . . . . . . 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 3305	. . . . . . 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 3105	. . . . . . 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 20731	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → 𝑦 ⊆ 𝑋)
37	36	sselda 3603	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝑋)
38		simprr2 1110	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑥 ∈ 𝑝)
39	5	ad3antrrr 766	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑋 = ∪ 𝐽)
40	39	difeq1d 3727	. . . . . . . . . . . . . . . . 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 766	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝐽 ∈ Top)
42		simprll 802	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑜 ∈ 𝐽)
43	9	opncld 20837	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑜) ∈ (Clsd‘𝐽))
44	41, 42, 43	syl2anc 693	. . . . . . . . . . . . . . . . 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 2701	. . . . . . . . . . . . . . . 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 3805	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∩ 𝑜) = (𝑜 ∩ 𝑝)
47		simprr3 1111	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → (𝑜 ∩ 𝑝) = ∅)
48	46, 47	syl5eq 2668	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → (𝑝 ∩ 𝑜) = ∅)
49		simplll 798	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝐽 ∈ (TopOn‘𝑋))
50		simprlr 803	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑝 ∈ 𝐽)
51		toponss 20731	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑝 ∈ 𝐽) → 𝑝 ⊆ 𝑋)
52	49, 50, 51	syl2anc 693	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑝 ⊆ 𝑋)
53		reldisj 4020	. . . . . . . . . . . . . . . . . 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 222	. . . . . . . . . . . . . . . 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 20876	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽) ∧ 𝑝 ⊆ (𝑋 ∖ 𝑜)) → ((cls‘𝐽)‘𝑝) ⊆ (𝑋 ∖ 𝑜))
57	45, 55, 56	syl2anc 693	. . . . . . . . . . . . . . 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 1109	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → (𝑋 ∖ 𝑦) ⊆ 𝑜)
59		difcom 4053	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑦)
60	58, 59	sylib 208	. . . . . . . . . . . . . . 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 3613	. . . . . . . . . . . . . 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 554	. . . . . . . . . . . . 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 643	. . . . . . . . . . . 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 680	. . . . . . . . . . 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 3017	. . . . . . . . . 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 3031	. . . . . . . . 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 2962	. . . . . . 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	syl5bi 232	. . . . . 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 2969	. . . 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 445	. . 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 21136	. . 3 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝐽 ∀𝑥 ∈ 𝑦 ∃𝑝 ∈ 𝐽 (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
74	15, 72, 73	sylanbrc 698	. 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 877	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  ∪ cuni 4436  ‘cfv 5888  Topctop 20698  TopOnctopon 20715  Clsdccld 20820  clsccl 20822  Regcreg 21113

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-iin 4523  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-top 20699  df-topon 20716  df-cld 20823  df-cls 20825  df-reg 21120

This theorem is referenced by: (None)