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Theorem mretopd 21131
Description: A Moore collection which is closed under finite unions called topological; such a collection is the closed sets of a canonically associated topology. (Contributed by Stefan O'Rear, 1-Feb-2015.)
Hypotheses
Ref Expression
mretopd.m (𝜑𝑀 ∈ (Moore‘𝐵))
mretopd.z (𝜑 → ∅ ∈ 𝑀)
mretopd.u ((𝜑𝑥𝑀𝑦𝑀) → (𝑥𝑦) ∈ 𝑀)
mretopd.j 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ 𝑀}
Assertion
Ref Expression
mretopd (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ 𝑀 = (Clsd‘𝐽)))
Distinct variable groups:   𝜑,𝑥,𝑦,𝑧   𝑥,𝑀,𝑦,𝑧   𝑥,𝐽,𝑦   𝑥,𝐵,𝑦,𝑧
Allowed substitution hint:   𝐽(𝑧)

Proof of Theorem mretopd
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 unieq 4649 . . . . . . . . 9 (𝑎 = ∅ → 𝑎 = ∅)
2 uni0 4670 . . . . . . . . 9 ∅ = ∅
31, 2syl6eq 2867 . . . . . . . 8 (𝑎 = ∅ → 𝑎 = ∅)
43eleq1d 2881 . . . . . . 7 (𝑎 = ∅ → ( 𝑎𝐽 ↔ ∅ ∈ 𝐽))
5 mretopd.j . . . . . . . . . . . . . 14 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ 𝑀}
6 ssrab2 3895 . . . . . . . . . . . . . 14 {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ 𝑀} ⊆ 𝒫 𝐵
75, 6eqsstri 3843 . . . . . . . . . . . . 13 𝐽 ⊆ 𝒫 𝐵
8 sstr 3817 . . . . . . . . . . . . 13 ((𝑎𝐽𝐽 ⊆ 𝒫 𝐵) → 𝑎 ⊆ 𝒫 𝐵)
97, 8mpan2 674 . . . . . . . . . . . 12 (𝑎𝐽𝑎 ⊆ 𝒫 𝐵)
109adantl 469 . . . . . . . . . . 11 ((𝜑𝑎𝐽) → 𝑎 ⊆ 𝒫 𝐵)
11 sspwuni 4814 . . . . . . . . . . 11 (𝑎 ⊆ 𝒫 𝐵 𝑎𝐵)
1210, 11sylib 209 . . . . . . . . . 10 ((𝜑𝑎𝐽) → 𝑎𝐵)
13 vuniex 7194 . . . . . . . . . . 11 𝑎 ∈ V
1413elpw 4368 . . . . . . . . . 10 ( 𝑎 ∈ 𝒫 𝐵 𝑎𝐵)
1512, 14sylibr 225 . . . . . . . . 9 ((𝜑𝑎𝐽) → 𝑎 ∈ 𝒫 𝐵)
1615adantr 468 . . . . . . . 8 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → 𝑎 ∈ 𝒫 𝐵)
17 uniiun 4776 . . . . . . . . . 10 𝑎 = 𝑏𝑎 𝑏
1817difeq2i 3935 . . . . . . . . 9 (𝐵 𝑎) = (𝐵 𝑏𝑎 𝑏)
19 iindif2 4792 . . . . . . . . . . 11 (𝑎 ≠ ∅ → 𝑏𝑎 (𝐵𝑏) = (𝐵 𝑏𝑎 𝑏))
2019adantl 469 . . . . . . . . . 10 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → 𝑏𝑎 (𝐵𝑏) = (𝐵 𝑏𝑎 𝑏))
21 mretopd.m . . . . . . . . . . . 12 (𝜑𝑀 ∈ (Moore‘𝐵))
2221ad2antrr 708 . . . . . . . . . . 11 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → 𝑀 ∈ (Moore‘𝐵))
23 simpr 473 . . . . . . . . . . 11 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
24 difeq2 3932 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑏 → (𝐵𝑧) = (𝐵𝑏))
2524eleq1d 2881 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑏 → ((𝐵𝑧) ∈ 𝑀 ↔ (𝐵𝑏) ∈ 𝑀))
2625, 5elrab2 3573 . . . . . . . . . . . . . . 15 (𝑏𝐽 ↔ (𝑏 ∈ 𝒫 𝐵 ∧ (𝐵𝑏) ∈ 𝑀))
2726simprbi 486 . . . . . . . . . . . . . 14 (𝑏𝐽 → (𝐵𝑏) ∈ 𝑀)
2827rgen 3121 . . . . . . . . . . . . 13 𝑏𝐽 (𝐵𝑏) ∈ 𝑀
29 ssralv 3874 . . . . . . . . . . . . . 14 (𝑎𝐽 → (∀𝑏𝐽 (𝐵𝑏) ∈ 𝑀 → ∀𝑏𝑎 (𝐵𝑏) ∈ 𝑀))
3029adantl 469 . . . . . . . . . . . . 13 ((𝜑𝑎𝐽) → (∀𝑏𝐽 (𝐵𝑏) ∈ 𝑀 → ∀𝑏𝑎 (𝐵𝑏) ∈ 𝑀))
3128, 30mpi 20 . . . . . . . . . . . 12 ((𝜑𝑎𝐽) → ∀𝑏𝑎 (𝐵𝑏) ∈ 𝑀)
3231adantr 468 . . . . . . . . . . 11 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → ∀𝑏𝑎 (𝐵𝑏) ∈ 𝑀)
33 mreiincl 16481 . . . . . . . . . . 11 ((𝑀 ∈ (Moore‘𝐵) ∧ 𝑎 ≠ ∅ ∧ ∀𝑏𝑎 (𝐵𝑏) ∈ 𝑀) → 𝑏𝑎 (𝐵𝑏) ∈ 𝑀)
3422, 23, 32, 33syl3anc 1483 . . . . . . . . . 10 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → 𝑏𝑎 (𝐵𝑏) ∈ 𝑀)
3520, 34eqeltrrd 2897 . . . . . . . . 9 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → (𝐵 𝑏𝑎 𝑏) ∈ 𝑀)
3618, 35syl5eqel 2900 . . . . . . . 8 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → (𝐵 𝑎) ∈ 𝑀)
37 difeq2 3932 . . . . . . . . . 10 (𝑧 = 𝑎 → (𝐵𝑧) = (𝐵 𝑎))
3837eleq1d 2881 . . . . . . . . 9 (𝑧 = 𝑎 → ((𝐵𝑧) ∈ 𝑀 ↔ (𝐵 𝑎) ∈ 𝑀))
3938, 5elrab2 3573 . . . . . . . 8 ( 𝑎𝐽 ↔ ( 𝑎 ∈ 𝒫 𝐵 ∧ (𝐵 𝑎) ∈ 𝑀))
4016, 36, 39sylanbrc 574 . . . . . . 7 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → 𝑎𝐽)
41 0elpw 5039 . . . . . . . . . 10 ∅ ∈ 𝒫 𝐵
4241a1i 11 . . . . . . . . 9 (𝜑 → ∅ ∈ 𝒫 𝐵)
43 mre1cl 16479 . . . . . . . . . 10 (𝑀 ∈ (Moore‘𝐵) → 𝐵𝑀)
4421, 43syl 17 . . . . . . . . 9 (𝜑𝐵𝑀)
45 difeq2 3932 . . . . . . . . . . . 12 (𝑧 = ∅ → (𝐵𝑧) = (𝐵 ∖ ∅))
46 dif0 4162 . . . . . . . . . . . 12 (𝐵 ∖ ∅) = 𝐵
4745, 46syl6eq 2867 . . . . . . . . . . 11 (𝑧 = ∅ → (𝐵𝑧) = 𝐵)
4847eleq1d 2881 . . . . . . . . . 10 (𝑧 = ∅ → ((𝐵𝑧) ∈ 𝑀𝐵𝑀))
4948, 5elrab2 3573 . . . . . . . . 9 (∅ ∈ 𝐽 ↔ (∅ ∈ 𝒫 𝐵𝐵𝑀))
5042, 44, 49sylanbrc 574 . . . . . . . 8 (𝜑 → ∅ ∈ 𝐽)
5150adantr 468 . . . . . . 7 ((𝜑𝑎𝐽) → ∅ ∈ 𝐽)
524, 40, 51pm2.61ne 3074 . . . . . 6 ((𝜑𝑎𝐽) → 𝑎𝐽)
5352ex 399 . . . . 5 (𝜑 → (𝑎𝐽 𝑎𝐽))
5453alrimiv 2018 . . . 4 (𝜑 → ∀𝑎(𝑎𝐽 𝑎𝐽))
55 inss1 4040 . . . . . . . 8 (𝑎𝑏) ⊆ 𝑎
56 difeq2 3932 . . . . . . . . . . . . 13 (𝑧 = 𝑎 → (𝐵𝑧) = (𝐵𝑎))
5756eleq1d 2881 . . . . . . . . . . . 12 (𝑧 = 𝑎 → ((𝐵𝑧) ∈ 𝑀 ↔ (𝐵𝑎) ∈ 𝑀))
5857, 5elrab2 3573 . . . . . . . . . . 11 (𝑎𝐽 ↔ (𝑎 ∈ 𝒫 𝐵 ∧ (𝐵𝑎) ∈ 𝑀))
5958simplbi 487 . . . . . . . . . 10 (𝑎𝐽𝑎 ∈ 𝒫 𝐵)
6059elpwid 4374 . . . . . . . . 9 (𝑎𝐽𝑎𝐵)
6160ad2antrl 710 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → 𝑎𝐵)
6255, 61syl5ss 3820 . . . . . . 7 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → (𝑎𝑏) ⊆ 𝐵)
63 vex 3405 . . . . . . . . 9 𝑎 ∈ V
6463inex1 5007 . . . . . . . 8 (𝑎𝑏) ∈ V
6564elpw 4368 . . . . . . 7 ((𝑎𝑏) ∈ 𝒫 𝐵 ↔ (𝑎𝑏) ⊆ 𝐵)
6662, 65sylibr 225 . . . . . 6 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → (𝑎𝑏) ∈ 𝒫 𝐵)
67 difindi 4094 . . . . . . 7 (𝐵 ∖ (𝑎𝑏)) = ((𝐵𝑎) ∪ (𝐵𝑏))
6858simprbi 486 . . . . . . . . 9 (𝑎𝐽 → (𝐵𝑎) ∈ 𝑀)
6968ad2antrl 710 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → (𝐵𝑎) ∈ 𝑀)
7027ad2antll 711 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → (𝐵𝑏) ∈ 𝑀)
71 simpl 470 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → 𝜑)
72 uneq1 3970 . . . . . . . . . . . 12 (𝑥 = (𝐵𝑎) → (𝑥𝑦) = ((𝐵𝑎) ∪ 𝑦))
7372eleq1d 2881 . . . . . . . . . . 11 (𝑥 = (𝐵𝑎) → ((𝑥𝑦) ∈ 𝑀 ↔ ((𝐵𝑎) ∪ 𝑦) ∈ 𝑀))
7473imbi2d 331 . . . . . . . . . 10 (𝑥 = (𝐵𝑎) → ((𝜑 → (𝑥𝑦) ∈ 𝑀) ↔ (𝜑 → ((𝐵𝑎) ∪ 𝑦) ∈ 𝑀)))
75 uneq2 3971 . . . . . . . . . . . 12 (𝑦 = (𝐵𝑏) → ((𝐵𝑎) ∪ 𝑦) = ((𝐵𝑎) ∪ (𝐵𝑏)))
7675eleq1d 2881 . . . . . . . . . . 11 (𝑦 = (𝐵𝑏) → (((𝐵𝑎) ∪ 𝑦) ∈ 𝑀 ↔ ((𝐵𝑎) ∪ (𝐵𝑏)) ∈ 𝑀))
7776imbi2d 331 . . . . . . . . . 10 (𝑦 = (𝐵𝑏) → ((𝜑 → ((𝐵𝑎) ∪ 𝑦) ∈ 𝑀) ↔ (𝜑 → ((𝐵𝑎) ∪ (𝐵𝑏)) ∈ 𝑀)))
78 mretopd.u . . . . . . . . . . . 12 ((𝜑𝑥𝑀𝑦𝑀) → (𝑥𝑦) ∈ 𝑀)
79783expb 1142 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑀𝑦𝑀)) → (𝑥𝑦) ∈ 𝑀)
8079expcom 400 . . . . . . . . . 10 ((𝑥𝑀𝑦𝑀) → (𝜑 → (𝑥𝑦) ∈ 𝑀))
8174, 77, 80vtocl2ga 3478 . . . . . . . . 9 (((𝐵𝑎) ∈ 𝑀 ∧ (𝐵𝑏) ∈ 𝑀) → (𝜑 → ((𝐵𝑎) ∪ (𝐵𝑏)) ∈ 𝑀))
8281imp 395 . . . . . . . 8 ((((𝐵𝑎) ∈ 𝑀 ∧ (𝐵𝑏) ∈ 𝑀) ∧ 𝜑) → ((𝐵𝑎) ∪ (𝐵𝑏)) ∈ 𝑀)
8369, 70, 71, 82syl21anc 857 . . . . . . 7 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → ((𝐵𝑎) ∪ (𝐵𝑏)) ∈ 𝑀)
8467, 83syl5eqel 2900 . . . . . 6 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → (𝐵 ∖ (𝑎𝑏)) ∈ 𝑀)
85 difeq2 3932 . . . . . . . 8 (𝑧 = (𝑎𝑏) → (𝐵𝑧) = (𝐵 ∖ (𝑎𝑏)))
8685eleq1d 2881 . . . . . . 7 (𝑧 = (𝑎𝑏) → ((𝐵𝑧) ∈ 𝑀 ↔ (𝐵 ∖ (𝑎𝑏)) ∈ 𝑀))
8786, 5elrab2 3573 . . . . . 6 ((𝑎𝑏) ∈ 𝐽 ↔ ((𝑎𝑏) ∈ 𝒫 𝐵 ∧ (𝐵 ∖ (𝑎𝑏)) ∈ 𝑀))
8866, 84, 87sylanbrc 574 . . . . 5 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → (𝑎𝑏) ∈ 𝐽)
8988ralrimivva 3170 . . . 4 (𝜑 → ∀𝑎𝐽𝑏𝐽 (𝑎𝑏) ∈ 𝐽)
9044pwexd 5062 . . . . . 6 (𝜑 → 𝒫 𝐵 ∈ V)
915, 90rabexd 5021 . . . . 5 (𝜑𝐽 ∈ V)
92 istopg 20934 . . . . 5 (𝐽 ∈ V → (𝐽 ∈ Top ↔ (∀𝑎(𝑎𝐽 𝑎𝐽) ∧ ∀𝑎𝐽𝑏𝐽 (𝑎𝑏) ∈ 𝐽)))
9391, 92syl 17 . . . 4 (𝜑 → (𝐽 ∈ Top ↔ (∀𝑎(𝑎𝐽 𝑎𝐽) ∧ ∀𝑎𝐽𝑏𝐽 (𝑎𝑏) ∈ 𝐽)))
9454, 89, 93mpbir2and 695 . . 3 (𝜑𝐽 ∈ Top)
957unissi 4666 . . . . . 6 𝐽 𝒫 𝐵
96 unipw 5121 . . . . . 6 𝒫 𝐵 = 𝐵
9795, 96sseqtri 3845 . . . . 5 𝐽𝐵
98 pwidg 4377 . . . . . . 7 (𝐵𝑀𝐵 ∈ 𝒫 𝐵)
9944, 98syl 17 . . . . . 6 (𝜑𝐵 ∈ 𝒫 𝐵)
100 difid 4160 . . . . . . 7 (𝐵𝐵) = ∅
101 mretopd.z . . . . . . 7 (𝜑 → ∅ ∈ 𝑀)
102100, 101syl5eqel 2900 . . . . . 6 (𝜑 → (𝐵𝐵) ∈ 𝑀)
103 difeq2 3932 . . . . . . . 8 (𝑧 = 𝐵 → (𝐵𝑧) = (𝐵𝐵))
104103eleq1d 2881 . . . . . . 7 (𝑧 = 𝐵 → ((𝐵𝑧) ∈ 𝑀 ↔ (𝐵𝐵) ∈ 𝑀))
105104, 5elrab2 3573 . . . . . 6 (𝐵𝐽 ↔ (𝐵 ∈ 𝒫 𝐵 ∧ (𝐵𝐵) ∈ 𝑀))
10699, 102, 105sylanbrc 574 . . . . 5 (𝜑𝐵𝐽)
107 unissel 4673 . . . . 5 (( 𝐽𝐵𝐵𝐽) → 𝐽 = 𝐵)
10897, 106, 107sylancr 577 . . . 4 (𝜑 𝐽 = 𝐵)
109108eqcomd 2823 . . 3 (𝜑𝐵 = 𝐽)
110 istopon 20951 . . 3 (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))
11194, 109, 110sylanbrc 574 . 2 (𝜑𝐽 ∈ (TopOn‘𝐵))
112 eqid 2817 . . . . 5 𝐽 = 𝐽
113112cldval 21062 . . . 4 (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝐽 ∣ ( 𝐽𝑥) ∈ 𝐽})
11494, 113syl 17 . . 3 (𝜑 → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝐽 ∣ ( 𝐽𝑥) ∈ 𝐽})
115108pweqd 4367 . . . 4 (𝜑 → 𝒫 𝐽 = 𝒫 𝐵)
116108difeq1d 3937 . . . . 5 (𝜑 → ( 𝐽𝑥) = (𝐵𝑥))
117116eleq1d 2881 . . . 4 (𝜑 → (( 𝐽𝑥) ∈ 𝐽 ↔ (𝐵𝑥) ∈ 𝐽))
118115, 117rabeqbidv 3396 . . 3 (𝜑 → {𝑥 ∈ 𝒫 𝐽 ∣ ( 𝐽𝑥) ∈ 𝐽} = {𝑥 ∈ 𝒫 𝐵 ∣ (𝐵𝑥) ∈ 𝐽})
1195eleq2i 2888 . . . . . . 7 ((𝐵𝑥) ∈ 𝐽 ↔ (𝐵𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ 𝑀})
120 difss 3947 . . . . . . . . . 10 (𝐵𝑥) ⊆ 𝐵
121 elpw2g 5032 . . . . . . . . . . 11 (𝐵𝑀 → ((𝐵𝑥) ∈ 𝒫 𝐵 ↔ (𝐵𝑥) ⊆ 𝐵))
12244, 121syl 17 . . . . . . . . . 10 (𝜑 → ((𝐵𝑥) ∈ 𝒫 𝐵 ↔ (𝐵𝑥) ⊆ 𝐵))
123120, 122mpbiri 249 . . . . . . . . 9 (𝜑 → (𝐵𝑥) ∈ 𝒫 𝐵)
124 difeq2 3932 . . . . . . . . . . 11 (𝑧 = (𝐵𝑥) → (𝐵𝑧) = (𝐵 ∖ (𝐵𝑥)))
125124eleq1d 2881 . . . . . . . . . 10 (𝑧 = (𝐵𝑥) → ((𝐵𝑧) ∈ 𝑀 ↔ (𝐵 ∖ (𝐵𝑥)) ∈ 𝑀))
126125elrab3 3571 . . . . . . . . 9 ((𝐵𝑥) ∈ 𝒫 𝐵 → ((𝐵𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ 𝑀} ↔ (𝐵 ∖ (𝐵𝑥)) ∈ 𝑀))
127123, 126syl 17 . . . . . . . 8 (𝜑 → ((𝐵𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ 𝑀} ↔ (𝐵 ∖ (𝐵𝑥)) ∈ 𝑀))
128127adantr 468 . . . . . . 7 ((𝜑𝑥 ∈ 𝒫 𝐵) → ((𝐵𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ 𝑀} ↔ (𝐵 ∖ (𝐵𝑥)) ∈ 𝑀))
129119, 128syl5bb 274 . . . . . 6 ((𝜑𝑥 ∈ 𝒫 𝐵) → ((𝐵𝑥) ∈ 𝐽 ↔ (𝐵 ∖ (𝐵𝑥)) ∈ 𝑀))
130 elpwi 4372 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
131 dfss4 4071 . . . . . . . . 9 (𝑥𝐵 ↔ (𝐵 ∖ (𝐵𝑥)) = 𝑥)
132130, 131sylib 209 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑥)) = 𝑥)
133132adantl 469 . . . . . . 7 ((𝜑𝑥 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵𝑥)) = 𝑥)
134133eleq1d 2881 . . . . . 6 ((𝜑𝑥 ∈ 𝒫 𝐵) → ((𝐵 ∖ (𝐵𝑥)) ∈ 𝑀𝑥𝑀))
135129, 134bitrd 270 . . . . 5 ((𝜑𝑥 ∈ 𝒫 𝐵) → ((𝐵𝑥) ∈ 𝐽𝑥𝑀))
136135rabbidva 3389 . . . 4 (𝜑 → {𝑥 ∈ 𝒫 𝐵 ∣ (𝐵𝑥) ∈ 𝐽} = {𝑥 ∈ 𝒫 𝐵𝑥𝑀})
137 incom 4015 . . . . . 6 (𝑀 ∩ 𝒫 𝐵) = (𝒫 𝐵𝑀)
138 dfin5 3788 . . . . . 6 (𝒫 𝐵𝑀) = {𝑥 ∈ 𝒫 𝐵𝑥𝑀}
139137, 138eqtri 2839 . . . . 5 (𝑀 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝒫 𝐵𝑥𝑀}
140 mresspw 16477 . . . . . . 7 (𝑀 ∈ (Moore‘𝐵) → 𝑀 ⊆ 𝒫 𝐵)
14121, 140syl 17 . . . . . 6 (𝜑𝑀 ⊆ 𝒫 𝐵)
142 df-ss 3794 . . . . . 6 (𝑀 ⊆ 𝒫 𝐵 ↔ (𝑀 ∩ 𝒫 𝐵) = 𝑀)
143141, 142sylib 209 . . . . 5 (𝜑 → (𝑀 ∩ 𝒫 𝐵) = 𝑀)
144139, 143syl5eqr 2865 . . . 4 (𝜑 → {𝑥 ∈ 𝒫 𝐵𝑥𝑀} = 𝑀)
145136, 144eqtrd 2851 . . 3 (𝜑 → {𝑥 ∈ 𝒫 𝐵 ∣ (𝐵𝑥) ∈ 𝐽} = 𝑀)
146114, 118, 1453eqtrrd 2856 . 2 (𝜑𝑀 = (Clsd‘𝐽))
147111, 146jca 503 1 (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ 𝑀 = (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100  wal 1635   = wceq 1637  wcel 2157  wne 2989  wral 3107  {crab 3111  Vcvv 3402  cdif 3777  cun 3778  cin 3779  wss 3780  c0 4127  𝒫 cpw 4362   cuni 4641   ciun 4723   ciin 4724  cfv 6111  Moorecmre 16467  Topctop 20932  TopOnctopon 20949  Clsdccld 21055
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-8 2159  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2795  ax-sep 4988  ax-nul 4996  ax-pow 5048  ax-pr 5109  ax-un 7189
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-mo 2635  df-eu 2642  df-clab 2804  df-cleq 2810  df-clel 2813  df-nfc 2948  df-ne 2990  df-ral 3112  df-rex 3113  df-rab 3116  df-v 3404  df-sbc 3645  df-dif 3783  df-un 3785  df-in 3787  df-ss 3794  df-nul 4128  df-if 4291  df-pw 4364  df-sn 4382  df-pr 4384  df-op 4388  df-uni 4642  df-int 4681  df-iun 4725  df-iin 4726  df-br 4856  df-opab 4918  df-mpt 4935  df-id 5232  df-xp 5330  df-rel 5331  df-cnv 5332  df-co 5333  df-dm 5334  df-iota 6074  df-fun 6113  df-fv 6119  df-mre 16471  df-top 20933  df-topon 20950  df-cld 21058
This theorem is referenced by:  iscldtop  21134  istopclsd  37783
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