MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isreg2 Structured version   Visualization version   GIF version

Theorem isreg2 22274
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 1200 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝐽 ∈ Reg)
2 simp2l 1201 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑐 ∈ (Clsd‘𝐽))
3 simp2r 1202 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑥𝑋)
4 simp1l 1199 . . . . . . 7 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝐽 ∈ (TopOn‘𝑋))
5 toponuni 21811 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
64, 5syl 17 . . . . . 6 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑋 = 𝐽)
73, 6eleqtrd 2840 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → 𝑥 𝐽)
8 simp3 1140 . . . . 5 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋) ∧ ¬ 𝑥𝑐) → ¬ 𝑥𝑐)
9 eqid 2737 . . . . . 6 𝐽 = 𝐽
109regsep2 22273 . . . . 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 1123 . . 3 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑐 ∈ (Clsd‘𝐽) ∧ 𝑥𝑋)) → (¬ 𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
1312ralrimivva 3112 . 2 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
14 topontop 21810 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top)
1514adantr 484 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝐽 ∈ Top)
165adantr 484 . . . . . . . . 9 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → 𝑋 = 𝐽)
1716difeq1d 4036 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (𝑋𝑦) = ( 𝐽𝑦))
189opncld 21930 . . . . . . . . 9 ((𝐽 ∈ Top ∧ 𝑦𝐽) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
1914, 18sylan 583 . . . . . . . 8 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → ( 𝐽𝑦) ∈ (Clsd‘𝐽))
2017, 19eqeltrd 2838 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (𝑋𝑦) ∈ (Clsd‘𝐽))
21 eleq2 2826 . . . . . . . . . . . 12 (𝑐 = (𝑋𝑦) → (𝑥𝑐𝑥 ∈ (𝑋𝑦)))
2221notbid 321 . . . . . . . . . . 11 (𝑐 = (𝑋𝑦) → (¬ 𝑥𝑐 ↔ ¬ 𝑥 ∈ (𝑋𝑦)))
23 eldif 3876 . . . . . . . . . . . . 13 (𝑥 ∈ (𝑋𝑦) ↔ (𝑥𝑋 ∧ ¬ 𝑥𝑦))
2423baibr 540 . . . . . . . . . . . 12 (𝑥𝑋 → (¬ 𝑥𝑦𝑥 ∈ (𝑋𝑦)))
2524con1bid 359 . . . . . . . . . . 11 (𝑥𝑋 → (¬ 𝑥 ∈ (𝑋𝑦) ↔ 𝑥𝑦))
2622, 25sylan9bb 513 . . . . . . . . . 10 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → (¬ 𝑥𝑐𝑥𝑦))
27 simpl 486 . . . . . . . . . . . . 13 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → 𝑐 = (𝑋𝑦))
2827sseq1d 3932 . . . . . . . . . . . 12 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → (𝑐𝑜 ↔ (𝑋𝑦) ⊆ 𝑜))
29283anbi1d 1442 . . . . . . . . . . 11 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → ((𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) ↔ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
30292rexbidv 3219 . . . . . . . . . 10 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → (∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) ↔ ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
3126, 30imbi12d 348 . . . . . . . . 9 ((𝑐 = (𝑋𝑦) ∧ 𝑥𝑋) → ((¬ 𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) ↔ (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))))
3231ralbidva 3117 . . . . . . . 8 (𝑐 = (𝑋𝑦) → (∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) ↔ ∀𝑥𝑋 (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))))
3332rspcv 3532 . . . . . . 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 3276 . . . . . . 7 (∀𝑥𝑋 (𝑥𝑦 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) ↔ ∀𝑥𝑦 (𝑥𝑋 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)))
36 toponss 21824 . . . . . . . . . 10 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → 𝑦𝑋)
3736sselda 3901 . . . . . . . . 9 (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) → 𝑥𝑋)
38 simprr2 1224 . . . . . . . . . . . . . 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 4036 . . . . . . . . . . . . . . . . 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 21930 . . . . . . . . . . . . . . . . . 18 ((𝐽 ∈ Top ∧ 𝑜𝐽) → ( 𝐽𝑜) ∈ (Clsd‘𝐽))
4441, 42, 43syl2anc 587 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ( 𝐽𝑜) ∈ (Clsd‘𝐽))
4540, 44eqeltrd 2838 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑜) ∈ (Clsd‘𝐽))
46 incom 4115 . . . . . . . . . . . . . . . . . 18 (𝑝𝑜) = (𝑜𝑝)
47 simprr3 1225 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑜𝑝) = ∅)
4846, 47syl5eq 2790 . . . . . . . . . . . . . . . . 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 21824 . . . . . . . . . . . . . . . . . . 19 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑝𝐽) → 𝑝𝑋)
5249, 50, 51syl2anc 587 . . . . . . . . . . . . . . . . . 18 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑝𝑋)
53 reldisj 4366 . . . . . . . . . . . . . . . . . 18 (𝑝𝑋 → ((𝑝𝑜) = ∅ ↔ 𝑝 ⊆ (𝑋𝑜)))
5452, 53syl 17 . . . . . . . . . . . . . . . . 17 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ((𝑝𝑜) = ∅ ↔ 𝑝 ⊆ (𝑋𝑜)))
5548, 54mpbid 235 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → 𝑝 ⊆ (𝑋𝑜))
569clsss2 21969 . . . . . . . . . . . . . . . 16 (((𝑋𝑜) ∈ (Clsd‘𝐽) ∧ 𝑝 ⊆ (𝑋𝑜)) → ((cls‘𝐽)‘𝑝) ⊆ (𝑋𝑜))
5745, 55, 56syl2anc 587 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ((cls‘𝐽)‘𝑝) ⊆ (𝑋𝑜))
58 simprr1 1223 . . . . . . . . . . . . . . . 16 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑦) ⊆ 𝑜)
59 difcom 4400 . . . . . . . . . . . . . . . 16 ((𝑋𝑦) ⊆ 𝑜 ↔ (𝑋𝑜) ⊆ 𝑦)
6058, 59sylib 221 . . . . . . . . . . . . . . 15 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑋𝑜) ⊆ 𝑦)
6157, 60sstrd 3911 . . . . . . . . . . . . . 14 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ((cls‘𝐽)‘𝑝) ⊆ 𝑦)
6238, 61jca 515 . . . . . . . . . . . . 13 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ ((𝑜𝐽𝑝𝐽) ∧ ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦))
6362expr 460 . . . . . . . . . . . 12 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ (𝑜𝐽𝑝𝐽)) → (((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) → (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6463anassrs 471 . . . . . . . . . . 11 (((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ 𝑜𝐽) ∧ 𝑝𝐽) → (((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) → (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6564reximdva 3193 . . . . . . . . . 10 ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) ∧ 𝑥𝑦) ∧ 𝑜𝐽) → (∃𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅) → ∃𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6665rexlimdva 3203 . . . . . . . . 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 3100 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑦𝐽) → (∀𝑥𝑦 (𝑥𝑋 → ∃𝑜𝐽𝑝𝐽 ((𝑋𝑦) ⊆ 𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) → ∀𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
6935, 68syl5bi 245 . . . . . 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 3110 . . . 4 (𝐽 ∈ (TopOn‘𝑋) → (∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅)) → ∀𝑦𝐽𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
7271imp 410 . . 3 ((𝐽 ∈ (TopOn‘𝑋) ∧ ∀𝑐 ∈ (Clsd‘𝐽)∀𝑥𝑋𝑥𝑐 → ∃𝑜𝐽𝑝𝐽 (𝑐𝑜𝑥𝑝 ∧ (𝑜𝑝) = ∅))) → ∀𝑦𝐽𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦))
73 isreg 22229 . . 3 (𝐽 ∈ Reg ↔ (𝐽 ∈ Top ∧ ∀𝑦𝐽𝑥𝑦𝑝𝐽 (𝑥𝑝 ∧ ((cls‘𝐽)‘𝑝) ⊆ 𝑦)))
7415, 72, 73sylanbrc 586 . 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 209  wa 399  w3a 1089   = wceq 1543  wcel 2110  wral 3061  wrex 3062  cdif 3863  cin 3865  wss 3866  c0 4237   cuni 4819  cfv 6380  Topctop 21790  TopOnctopon 21807  Clsdccld 21913  clsccl 21915  Regcreg 22206
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-rep 5179  ax-sep 5192  ax-nul 5199  ax-pow 5258  ax-pr 5322  ax-un 7523
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-sbc 3695  df-csb 3812  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-pw 4515  df-sn 4542  df-pr 4544  df-op 4548  df-uni 4820  df-int 4860  df-iun 4906  df-iin 4907  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-res 5563  df-ima 5564  df-iota 6338  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387  df-fv 6388  df-top 21791  df-topon 21808  df-cld 21916  df-cls 21918  df-reg 22213
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator