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Theorem isreg2 21229
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 1106 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝐽 ∈ Reg)
2 simp2l 1107 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑐 ∈ (Clsd‘𝐽))
3 simp2r 1108 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑥𝑋)
4 simp1l 1105 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝐽 ∈ (TopOn‘𝑋))
5 toponuni 20767 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
64, 5syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑋 = 𝐽)
73, 6eleqtrd 2732 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑥 𝐽)
8 simp3 1083 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → ¬ 𝑥𝑐)
9 eqid 2651 . . . . . 6 𝐽 = 𝐽
109regsep2 21228 . . . . 5 ((𝐽 ∈ Reg ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥 𝐽 ∧ ¬ 𝑥𝑐)) → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))
111, 2, 7, 8, 10syl13anc 1368 . . . 4 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))
12113expia 1286 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋)) → (¬ 𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
1312ralrimivva 3000 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
14 topontop 20766 . . . 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 3760 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (𝑋𝑦) = ( 𝐽𝑦))
189opncld 20885 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
1914, 18sylan 487 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
2017, 19eqeltrd 2730 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (𝑋𝑦) ∈ (Clsd‘𝐽))
21 eleq2 2719 . . . . . . . . . . . 12 (𝑐 = (𝑋𝑦) → (𝑥𝑐𝑥 ∈ (𝑋𝑦)))
2221notbid 307 . . . . . . . . . . 11 (𝑐 = (𝑋𝑦) → (¬ 𝑥𝑐 ↔ ¬ 𝑥 ∈ (𝑋𝑦)))
23 eldif 3617 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑋𝑦) ↔ (𝑥𝑋 ∧ ¬ 𝑥𝑦))
2423baibr 965 . . . . . . . . . . . 12 (𝑥𝑋 → (¬ 𝑥𝑦𝑥 ∈ (𝑋𝑦)))
2524con1bid 344 . . . . . . . . . . 11 (𝑥𝑋 → (¬ 𝑥 ∈ (𝑋𝑦) ↔ 𝑥𝑦))
2622, 25sylan9bb 736 . . . . . . . . . 10 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → (¬ 𝑥𝑐𝑥𝑦))
27 simpl 472 . . . . . . . . . . . . 13 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → 𝑐 = (𝑋𝑦))
2827sseq1d 3665 . . . . . . . . . . . 12 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → (𝑐𝑜 ↔ (𝑋𝑦) ⊆ 𝑜))
29283anbi1d 1443 . . . . . . . . . . 11 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → ((𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) ↔ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
30292rexbidv 3086 . . . . . . . . . 10 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → (∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) ↔ ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
3126, 30imbi12d 333 . . . . . . . . 9 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → ((¬ 𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) ↔ (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))))
3231ralbidva 3014 . . . . . . . 8 (𝑐 = (𝑋𝑦) → (∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) ↔ ∀𝑥𝑋 (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))))
3332rspcv 3336 . . . . . . 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 3134 . . . . . . 7 (∀𝑥𝑋 (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) ↔ ∀𝑥𝑦 (𝑥𝑋 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
36 toponss 20779 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → 𝑦𝑋)
3736sselda 3636 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) → 𝑥𝑋)
38 simprr2 1130 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑥𝑝)
395ad3antrrr 766 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑋 = 𝐽)
4039difeq1d 3760 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑜) = ( 𝐽𝑜))
4114ad3antrrr 766 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝐽 ∈ Top)
42 simprll 819 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑜𝐽)
439opncld 20885 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ( 𝐽𝑜) ∈ (Clsd‘𝐽))
4441, 42, 43syl2anc 694 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ( 𝐽𝑜) ∈ (Clsd‘𝐽))
4540, 44eqeltrd 2730 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑜) ∈ (Clsd‘𝐽))
46 incom 3838 . . . . . . . . . . . . . . . . . 18 (𝑝𝑜) = (𝑜𝑝)
47 simprr3 1131 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑜𝑝) = ∅)
4846, 47syl5eq 2697 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑝𝑜) = ∅)
49 simplll 813 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝐽 ∈ (TopOn‘𝑋))
50 simprlr 820 . . . . . . . . . . . . . . . . . . 19 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑝𝐽)
51 toponss 20779 . . . . . . . . . . . . . . . . . . 19 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑝𝐽) → 𝑝𝑋)
5249, 50, 51syl2anc 694 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑝𝑋)
53 reldisj 4053 . . . . . . . . . . . . . . . . . 18 (𝑝𝑋 → ((𝑝𝑜) = ∅ ↔ 𝑝 ⊆ (𝑋𝑜)))
5452, 53syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ((𝑝𝑜) = ∅ ↔ 𝑝 ⊆ (𝑋𝑜)))
5548, 54mpbid 222 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑝 ⊆ (𝑋𝑜))
569clsss2 20924 . . . . . . . . . . . . . . . 16 (((𝑋𝑜) ∈ (Clsd‘𝐽) ∧ 𝑝 ⊆ (𝑋𝑜)) → ((cls‘𝐽)‘𝑝) ⊆ (𝑋𝑜))
5745, 55, 56syl2anc 694 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ((cls‘𝐽)‘𝑝) ⊆ (𝑋𝑜))
58 simprr1 1129 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑦) ⊆ 𝑜)
59 difcom 4086 . . . . . . . . . . . . . . . 16 ((𝑋𝑦) ⊆ 𝑜 ↔ (𝑋𝑜) ⊆ 𝑦)
6058, 59sylib 208 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑜) ⊆ 𝑦)
6157, 60sstrd 3646 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ((cls‘𝐽)‘𝑝) ⊆ 𝑦)
6238, 61jca 553 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦))
6362expr 642 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ (𝑜𝐽𝑝𝐽)) → (((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) → (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6463anassrs 681 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ 𝑜𝐽) ∧ 𝑝𝐽) → (((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) → (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6564reximdva 3046 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ 𝑜𝐽) → (∃𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) → ∃𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6665rexlimdva 3060 . . . . . . . . 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 2991 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (∀𝑥𝑦 (𝑥𝑋 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) → ∀𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6935, 68syl5bi 232 . . . . . 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 2998 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) → ∀𝑦𝐽𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
7271imp 444 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ∀𝑦𝐽𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦))
73 isreg 21184 . . 3 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑦𝐽𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
7415, 72, 73sylanbrc 699 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝐽 ∈ Reg)
7513, 74impbida 895 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wral 2941  wrex 2942  cdif 3604  cin 3606  wss 3607  c0 3948   cuni 4468  cfv 5926  Topctop 20746  TopOnctopon 20763  Clsdccld 20868  clsccl 20870  Regcreg 21161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-iin 4555  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-top 20747  df-topon 20764  df-cld 20871  df-cls 20873  df-reg 21168
This theorem is referenced by: (None)
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