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Theorem isreg2 21913
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 1190 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝐽 ∈ Reg)
2 simp2l 1191 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑐 ∈ (Clsd‘𝐽))
3 simp2r 1192 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑥𝑋)
4 simp1l 1189 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝐽 ∈ (TopOn‘𝑋))
5 toponuni 21450 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
64, 5syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑋 = 𝐽)
73, 6eleqtrd 2912 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑥 𝐽)
8 simp3 1130 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → ¬ 𝑥𝑐)
9 eqid 2818 . . . . . 6 𝐽 = 𝐽
109regsep2 21912 . . . . 5 ((𝐽 ∈ Reg ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 𝐽 ∧ ¬ 𝑥𝑐)) → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))
111, 2, 7, 8, 10syl13anc 1364 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))
12113expia 1113 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋)) → (¬ 𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
1312ralrimivva 3188 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
14 topontop 21449 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1514adantr 481 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝐽 ∈ Top)
165adantr 481 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → 𝑋 = 𝐽)
1716difeq1d 4095 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (𝑋𝑦) = ( 𝐽𝑦))
189opncld 21569 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
1914, 18sylan 580 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
2017, 19eqeltrd 2910 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (𝑋𝑦) ∈ (Clsd‘𝐽))
21 eleq2 2898 . . . . . . . . . . . 12 (𝑐 = (𝑋𝑦) → (𝑥𝑐𝑥 ∈ (𝑋𝑦)))
2221notbid 319 . . . . . . . . . . 11 (𝑐 = (𝑋𝑦) → (¬ 𝑥𝑐 ↔ ¬ 𝑥 ∈ (𝑋𝑦)))
23 eldif 3943 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑋𝑦) ↔ (𝑥𝑋 ∧ ¬ 𝑥𝑦))
2423baibr 537 . . . . . . . . . . . 12 (𝑥𝑋 → (¬ 𝑥𝑦𝑥 ∈ (𝑋𝑦)))
2524con1bid 357 . . . . . . . . . . 11 (𝑥𝑋 → (¬ 𝑥 ∈ (𝑋𝑦) ↔ 𝑥𝑦))
2622, 25sylan9bb 510 . . . . . . . . . 10 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → (¬ 𝑥𝑐𝑥𝑦))
27 simpl 483 . . . . . . . . . . . . 13 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → 𝑐 = (𝑋𝑦))
2827sseq1d 3995 . . . . . . . . . . . 12 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → (𝑐𝑜 ↔ (𝑋𝑦) ⊆ 𝑜))
29283anbi1d 1431 . . . . . . . . . . 11 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → ((𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) ↔ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
30292rexbidv 3297 . . . . . . . . . 10 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → (∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) ↔ ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
3126, 30imbi12d 346 . . . . . . . . 9 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → ((¬ 𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) ↔ (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))))
3231ralbidva 3193 . . . . . . . 8 (𝑐 = (𝑋𝑦) → (∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) ↔ ∀𝑥𝑋 (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))))
3332rspcv 3615 . . . . . . 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 3362 . . . . . . 7 (∀𝑥𝑋 (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) ↔ ∀𝑥𝑦 (𝑥𝑋 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
36 toponss 21463 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → 𝑦𝑋)
3736sselda 3964 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) → 𝑥𝑋)
38 simprr2 1214 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑥𝑝)
395ad3antrrr 726 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑋 = 𝐽)
4039difeq1d 4095 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑜) = ( 𝐽𝑜))
4114ad3antrrr 726 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝐽 ∈ Top)
42 simprll 775 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑜𝐽)
439opncld 21569 . . . . . . . . . . . . . . . . . 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 2910 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑜) ∈ (Clsd‘𝐽))
46 incom 4175 . . . . . . . . . . . . . . . . . 18 (𝑝𝑜) = (𝑜𝑝)
47 simprr3 1215 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑜𝑝) = ∅)
4846, 47syl5eq 2865 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑝𝑜) = ∅)
49 simplll 771 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝐽 ∈ (TopOn‘𝑋))
50 simprlr 776 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑝𝐽)
51 toponss 21463 . . . . . . . . . . . . . . . . . . 19 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑝𝐽) → 𝑝𝑋)
5249, 50, 51syl2anc 584 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑝𝑋)
53 reldisj 4398 . . . . . . . . . . . . . . . . . 18 (𝑝𝑋 → ((𝑝𝑜) = ∅ ↔ 𝑝 ⊆ (𝑋𝑜)))
5452, 53syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ((𝑝𝑜) = ∅ ↔ 𝑝 ⊆ (𝑋𝑜)))
5548, 54mpbid 233 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑝 ⊆ (𝑋𝑜))
569clsss2 21608 . . . . . . . . . . . . . . . 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 1213 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑦) ⊆ 𝑜)
59 difcom 4430 . . . . . . . . . . . . . . . 16 ((𝑋𝑦) ⊆ 𝑜 ↔ (𝑋𝑜) ⊆ 𝑦)
6058, 59sylib 219 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑜) ⊆ 𝑦)
6157, 60sstrd 3974 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ((cls‘𝐽)‘𝑝) ⊆ 𝑦)
6238, 61jca 512 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦))
6362expr 457 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ (𝑜𝐽𝑝𝐽)) → (((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) → (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6463anassrs 468 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ 𝑜𝐽) ∧ 𝑝𝐽) → (((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) → (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6564reximdva 3271 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ 𝑜𝐽) → (∃𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) → ∃𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6665rexlimdva 3281 . . . . . . . . 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 3174 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (∀𝑥𝑦 (𝑥𝑋 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) → ∀𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6935, 68syl5bi 243 . . . . . 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 3186 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) → ∀𝑦𝐽𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
7271imp 407 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ∀𝑦𝐽𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦))
73 isreg 21868 . . 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 797 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 207  wa 396  w3a 1079   = wceq 1528  wcel 2105  wral 3135  wrex 3136  cdif 3930  cin 3932  wss 3933  c0 4288   cuni 4830  cfv 6348  Topctop 21429  TopOnctopon 21446  Clsdccld 21552  clsccl 21554  Regcreg 21845
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-int 4868  df-iun 4912  df-iin 4913  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-top 21430  df-topon 21447  df-cld 21555  df-cls 21557  df-reg 21852
This theorem is referenced by: (None)
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