Theorem isreg2 23700

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 1217	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝐽 ∈ Reg)
2		simp2l 1218	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝑐 ∈ (Clsd‘𝐽))
3		simp2r 1219	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝑥 ∈ 𝑋)
4		simp1l 1216	. . . . . . 7 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝐽 ∈ (TopOn‘𝑋))
5		toponuni 23237	. . . . . . 7 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽)
6	4, 5	syl 18	. . . . . 6 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝑋 = ∪ 𝐽)
7	3, 6	eleqtrd 2862	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → 𝑥 ∈ ∪ 𝐽)
8		simp3 1156	. . . . 5 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 ∈ 𝑋) ∧ ¬ 𝑥 ∈ 𝑐) → ¬ 𝑥 ∈ 𝑐)
9		eqid 2760	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
10	9	regsep2 23699	. . . . 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 1399	. . . 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 1139	. . 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 3205	. 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 23236	. . . 4 ⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
15	14	adantr 486	. . 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 486	. . . . . . . . 9 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → 𝑋 = ∪ 𝐽)
17	16	difeq1d 4062	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (𝑋 ∖ 𝑦) = (∪ 𝐽 ∖ 𝑦))
18	9	opncld 23356	. . . . . . . . 9 ⊢ ((𝐽 ∈ Top ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
19	14, 18	sylan 592	. . . . . . . 8 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑦) ∈ (Clsd‘𝐽))
20	17, 19	eqeltrd 2860	. . . . . . 7 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → (𝑋 ∖ 𝑦) ∈ (Clsd‘𝐽))
21		eleq2 2849	. . . . . . . . . . . 12 ⊢ (𝑐 = (𝑋 ∖ 𝑦) → (𝑥 ∈ 𝑐 ↔ 𝑥 ∈ (𝑋 ∖ 𝑦)))
22	21	notbid 321	. . . . . . . . . . 11 ⊢ (𝑐 = (𝑋 ∖ 𝑦) → (¬ 𝑥 ∈ 𝑐 ↔ ¬ 𝑥 ∈ (𝑋 ∖ 𝑦)))
23		eldif 4049	. . . . . . . . . . . . 13 ⊢ (𝑥 ∈ (𝑋 ∖ 𝑦) ↔ (𝑥 ∈ 𝑋 ∧ ¬ 𝑥 ∈ 𝑦))
24	23	baibr 546	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ 𝑦 ↔ 𝑥 ∈ (𝑋 ∖ 𝑦)))
25	24	con1bid 358	. . . . . . . . . . 11 ⊢ (𝑥 ∈ 𝑋 → (¬ 𝑥 ∈ (𝑋 ∖ 𝑦) ↔ 𝑥 ∈ 𝑦))
26	22, 25	sylan9bb 519	. . . . . . . . . 10 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → (¬ 𝑥 ∈ 𝑐 ↔ 𝑥 ∈ 𝑦))
27		simpl 488	. . . . . . . . . . . . 13 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → 𝑐 = (𝑋 ∖ 𝑦))
28	27	sseq1d 3939	. . . . . . . . . . . 12 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → (𝑐 ⊆ 𝑜 ↔ (𝑋 ∖ 𝑦) ⊆ 𝑜))
29	28	3anbi1d 1468	. . . . . . . . . . 11 ⊢ ((𝑐 = (𝑋 ∖ 𝑦) ∧ 𝑥 ∈ 𝑋) → ((𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅) ↔ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅)))
30	29	2rexbidv 3227	. . . . . . . . . 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 347	. . . . . . . . 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 3183	. . . . . . . 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 3572	. . . . . . 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 18	. . . . . 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 3112	. . . . . . 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 23250	. . . . . . . . . 10 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) → 𝑦 ⊆ 𝑋)
37	36	sselda 3908	. . . . . . . . 9 ⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) → 𝑥 ∈ 𝑋)
38		simprr2 1241	. . . . . . . . . . . . . 14 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑥 ∈ 𝑝)
39	5	ad3antrrr 743	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑋 = ∪ 𝐽)
40	39	difeq1d 4062	. . . . . . . . . . . . . . . . 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 743	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝐽 ∈ Top)
42		simprll 791	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑜 ∈ 𝐽)
43	9	opncld 23356	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐽 ∈ Top ∧ 𝑜 ∈ 𝐽) → (∪ 𝐽 ∖ 𝑜) ∈ (Clsd‘𝐽))
44	41, 42, 43	syl2anc 596	. . . . . . . . . . . . . . . . 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 2860	. . . . . . . . . . . . . . . 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 4106	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ∩ 𝑜) = (𝑜 ∩ 𝑝)
47		simprr3 1242	. . . . . . . . . . . . . . . . . 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 2807	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → (𝑝 ∩ 𝑜) = ∅)
49		simplll 787	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝐽 ∈ (TopOn‘𝑋))
50		simprlr 792	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑝 ∈ 𝐽)
51		toponss 23250	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑝 ∈ 𝐽) → 𝑝 ⊆ 𝑋)
52	49, 50, 51	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → 𝑝 ⊆ 𝑋)
53		reldisj 4405	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑝 ⊆ 𝑋 → ((𝑝 ∩ 𝑜) = ∅ ↔ 𝑝 ⊆ (𝑋 ∖ 𝑜)))
54	52, 53	syl 18	. . . . . . . . . . . . . . . . 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 235	. . . . . . . . . . . . . . . 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 23395	. . . . . . . . . . . . . . . 16 ⊢ (((𝑋 ∖ 𝑜) ∈ (Clsd‘𝐽) ∧ 𝑝 ⊆ (𝑋 ∖ 𝑜)) → ((cls‘𝐽)‘𝑝) ⊆ (𝑋 ∖ 𝑜))
57	45, 55, 56	syl2anc 596	. . . . . . . . . . . . . . 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 1240	. . . . . . . . . . . . . . . 16 ⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝐽) ∧ 𝑥 ∈ 𝑦) ∧ ((𝑜 ∈ 𝐽 ∧ 𝑝 ∈ 𝐽) ∧ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))) → (𝑋 ∖ 𝑦) ⊆ 𝑜)
59		difcom 4443	. . . . . . . . . . . . . . . 16 ⊢ ((𝑋 ∖ 𝑦) ⊆ 𝑜 ↔ (𝑋 ∖ 𝑜) ⊆ 𝑦)
60	58, 59	sylib 221	. . . . . . . . . . . . . . 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 3918	. . . . . . . . . . . . . 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 521	. . . . . . . . . . . . 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 462	. . . . . . . . . . . 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 473	. . . . . . . . . . 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 3175	. . . . . . . . . 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 3163	. . . . . . . . 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 60	. . . . . . . 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 3174	. . . . . . 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 245	. . . . . 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 48	. . . . 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 3162	. . . 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 412	. . 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 23655	. . 3 ⊢ (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑦 ∈ 𝐽 ∀𝑥 ∈ 𝑦 ∃𝑝 ∈ 𝐽 (𝑥 ∈ 𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
74	15, 72, 73	sylanbrc 595	. 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 813	1 ⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Reg ↔ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥 ∈ 𝑋 (¬ 𝑥 ∈ 𝑐 → ∃𝑜 ∈ 𝐽 ∃𝑝 ∈ 𝐽 (𝑐 ⊆ 𝑜 ∧ 𝑥 ∈ 𝑝 ∧ (𝑜 ∩ 𝑝) = ∅))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569   ∈ wcel 2144  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∖ cdif 4045   ∩ cin 4089  ∅c0 4278  ∪ cuni 4866  ‘cfv 6538  Topctop 23216  TopOnctopon 23233  Clsdccld 23339  clsccl 23341  Regcreg 23632

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-int 4907  df-iun 4952  df-iin 4953  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-top 23217  df-topon 23234  df-cld 23342  df-cls 23344  df-reg 23639

This theorem is used by: (None)