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Theorem isreg2 23385
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)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))))
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.
StepHypRef Expression
1 simp1r 1199 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝐽 ∈ Reg)
2 simp2l 1200 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑐 ∈ (Clsd‘𝐽))
3 simp2r 1201 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑥𝑋)
4 simp1l 1198 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝐽 ∈ (TopOn‘𝑋))
5 toponuni 22920 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
64, 5syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑋 = 𝐽)
73, 6eleqtrd 2843 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑥 𝐽)
8 simp3 1139 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → ¬ 𝑥𝑐)
9 eqid 2737 . . . . . 6 𝐽 = 𝐽
109regsep2 23384 . . . . 5 ((𝐽 ∈ Reg ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 𝐽 ∧ ¬ 𝑥𝑐)) → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))
111, 2, 7, 8, 10syl13anc 1374 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))
12113expia 1122 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋)) → (¬ 𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
1312ralrimivva 3202 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
14 topontop 22919 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1514adantr 480 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝐽 ∈ Top)
165adantr 480 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → 𝑋 = 𝐽)
1716difeq1d 4125 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (𝑋𝑦) = ( 𝐽𝑦))
189opncld 23041 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
1914, 18sylan 580 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
2017, 19eqeltrd 2841 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (𝑋𝑦) ∈ (Clsd‘𝐽))
21 eleq2 2830 . . . . . . . . . . . 12 (𝑐 = (𝑋𝑦) → (𝑥𝑐𝑥 ∈ (𝑋𝑦)))
2221notbid 318 . . . . . . . . . . 11 (𝑐 = (𝑋𝑦) → (¬ 𝑥𝑐 ↔ ¬ 𝑥 ∈ (𝑋𝑦)))
23 eldif 3961 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑋𝑦) ↔ (𝑥𝑋 ∧ ¬ 𝑥𝑦))
2423baibr 536 . . . . . . . . . . . 12 (𝑥𝑋 → (¬ 𝑥𝑦𝑥 ∈ (𝑋𝑦)))
2524con1bid 355 . . . . . . . . . . 11 (𝑥𝑋 → (¬ 𝑥 ∈ (𝑋𝑦) ↔ 𝑥𝑦))
2622, 25sylan9bb 509 . . . . . . . . . 10 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → (¬ 𝑥𝑐𝑥𝑦))
27 simpl 482 . . . . . . . . . . . . 13 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → 𝑐 = (𝑋𝑦))
2827sseq1d 4015 . . . . . . . . . . . 12 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → (𝑐𝑜 ↔ (𝑋𝑦) ⊆ 𝑜))
29283anbi1d 1442 . . . . . . . . . . 11 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → ((𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) ↔ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
30292rexbidv 3222 . . . . . . . . . 10 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → (∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) ↔ ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
3126, 30imbi12d 344 . . . . . . . . 9 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → ((¬ 𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) ↔ (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))))
3231ralbidva 3176 . . . . . . . 8 (𝑐 = (𝑋𝑦) → (∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) ↔ ∀𝑥𝑋 (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))))
3332rspcv 3618 . . . . . . 7 ((𝑋𝑦) ∈ (Clsd‘𝐽) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) → ∀𝑥𝑋 (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))))
3420, 33syl 17 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) → ∀𝑥𝑋 (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))))
35 ralcom3 3097 . . . . . . 7 (∀𝑥𝑋 (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) ↔ ∀𝑥𝑦 (𝑥𝑋 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
36 toponss 22933 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → 𝑦𝑋)
3736sselda 3983 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) → 𝑥𝑋)
38 simprr2 1223 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑥𝑝)
395ad3antrrr 730 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑋 = 𝐽)
4039difeq1d 4125 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑜) = ( 𝐽𝑜))
4114ad3antrrr 730 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝐽 ∈ Top)
42 simprll 779 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑜𝐽)
439opncld 23041 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ( 𝐽𝑜) ∈ (Clsd‘𝐽))
4441, 42, 43syl2anc 584 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ( 𝐽𝑜) ∈ (Clsd‘𝐽))
4540, 44eqeltrd 2841 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑜) ∈ (Clsd‘𝐽))
46 incom 4209 . . . . . . . . . . . . . . . . . 18 (𝑝𝑜) = (𝑜𝑝)
47 simprr3 1224 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑜𝑝) = ∅)
4846, 47eqtrid 2789 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑝𝑜) = ∅)
49 simplll 775 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝐽 ∈ (TopOn‘𝑋))
50 simprlr 780 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑝𝐽)
51 toponss 22933 . . . . . . . . . . . . . . . . . . 19 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑝𝐽) → 𝑝𝑋)
5249, 50, 51syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑝𝑋)
53 reldisj 4453 . . . . . . . . . . . . . . . . . 18 (𝑝𝑋 → ((𝑝𝑜) = ∅ ↔ 𝑝 ⊆ (𝑋𝑜)))
5452, 53syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ((𝑝𝑜) = ∅ ↔ 𝑝 ⊆ (𝑋𝑜)))
5548, 54mpbid 232 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑝 ⊆ (𝑋𝑜))
569clsss2 23080 . . . . . . . . . . . . . . . 16 (((𝑋𝑜) ∈ (Clsd‘𝐽) ∧ 𝑝 ⊆ (𝑋𝑜)) → ((cls‘𝐽)‘𝑝) ⊆ (𝑋𝑜))
5745, 55, 56syl2anc 584 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ((cls‘𝐽)‘𝑝) ⊆ (𝑋𝑜))
58 simprr1 1222 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑦) ⊆ 𝑜)
59 difcom 4489 . . . . . . . . . . . . . . . 16 ((𝑋𝑦) ⊆ 𝑜 ↔ (𝑋𝑜) ⊆ 𝑦)
6058, 59sylib 218 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑜) ⊆ 𝑦)
6157, 60sstrd 3994 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ((cls‘𝐽)‘𝑝) ⊆ 𝑦)
6238, 61jca 511 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦))
6362expr 456 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ (𝑜𝐽𝑝𝐽)) → (((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) → (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6463anassrs 467 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ 𝑜𝐽) ∧ 𝑝𝐽) → (((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) → (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6564reximdva 3168 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ 𝑜𝐽) → (∃𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) → ∃𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6665rexlimdva 3155 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) → (∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) → ∃𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6737, 66embantd 59 . . . . . . . 8 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) → ((𝑥𝑋 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) → ∃𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6867ralimdva 3167 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (∀𝑥𝑦 (𝑥𝑋 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) → ∀𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6935, 68biimtrid 242 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (∀𝑥𝑋 (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) → ∀𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
7034, 69syld 47 . . . . 5 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) → ∀𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
7170ralrimdva 3154 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) → ∀𝑦𝐽𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
7271imp 406 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ∀𝑦𝐽𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦))
73 isreg 23340 . . 3 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑦𝐽𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
7415, 72, 73sylanbrc 583 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝐽 ∈ Reg)
7513, 74impbida 801 1 (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Reg ↔ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3a 1087   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cdif 3948  cin 3950  wss 3951  c0 4333   cuni 4907  cfv 6561  Topctop 22899  TopOnctopon 22916  Clsdccld 23024  clsccl 23026  Regcreg 23317
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 2708  ax-rep 5279  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-iun 4993  df-iin 4994  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-top 22900  df-topon 22917  df-cld 23027  df-cls 23029  df-reg 23324
This theorem is referenced by: (None)
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