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Theorem mretopd 23115
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 4922 . . . . . . . . 9 (𝑎 = ∅ → 𝑎 = ∅)
2 uni0 4939 . . . . . . . . 9 ∅ = ∅
31, 2eqtrdi 2790 . . . . . . . 8 (𝑎 = ∅ → 𝑎 = ∅)
43eleq1d 2823 . . . . . . 7 (𝑎 = ∅ → ( 𝑎𝐽 ↔ ∅ ∈ 𝐽))
5 mretopd.j . . . . . . . . . . . . . 14 𝐽 = {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ 𝑀}
65ssrab3 4091 . . . . . . . . . . . . 13 𝐽 ⊆ 𝒫 𝐵
7 sstr 4003 . . . . . . . . . . . . 13 ((𝑎𝐽𝐽 ⊆ 𝒫 𝐵) → 𝑎 ⊆ 𝒫 𝐵)
86, 7mpan2 691 . . . . . . . . . . . 12 (𝑎𝐽𝑎 ⊆ 𝒫 𝐵)
98adantl 481 . . . . . . . . . . 11 ((𝜑𝑎𝐽) → 𝑎 ⊆ 𝒫 𝐵)
10 sspwuni 5104 . . . . . . . . . . 11 (𝑎 ⊆ 𝒫 𝐵 𝑎𝐵)
119, 10sylib 218 . . . . . . . . . 10 ((𝜑𝑎𝐽) → 𝑎𝐵)
12 vuniex 7757 . . . . . . . . . . 11 𝑎 ∈ V
1312elpw 4608 . . . . . . . . . 10 ( 𝑎 ∈ 𝒫 𝐵 𝑎𝐵)
1411, 13sylibr 234 . . . . . . . . 9 ((𝜑𝑎𝐽) → 𝑎 ∈ 𝒫 𝐵)
1514adantr 480 . . . . . . . 8 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → 𝑎 ∈ 𝒫 𝐵)
16 uniiun 5062 . . . . . . . . . 10 𝑎 = 𝑏𝑎 𝑏
1716difeq2i 4132 . . . . . . . . 9 (𝐵 𝑎) = (𝐵 𝑏𝑎 𝑏)
18 iindif2 5081 . . . . . . . . . . 11 (𝑎 ≠ ∅ → 𝑏𝑎 (𝐵𝑏) = (𝐵 𝑏𝑎 𝑏))
1918adantl 481 . . . . . . . . . 10 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → 𝑏𝑎 (𝐵𝑏) = (𝐵 𝑏𝑎 𝑏))
20 mretopd.m . . . . . . . . . . . 12 (𝜑𝑀 ∈ (Moore‘𝐵))
2120ad2antrr 726 . . . . . . . . . . 11 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → 𝑀 ∈ (Moore‘𝐵))
22 simpr 484 . . . . . . . . . . 11 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → 𝑎 ≠ ∅)
23 difeq2 4129 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑏 → (𝐵𝑧) = (𝐵𝑏))
2423eleq1d 2823 . . . . . . . . . . . . . . . 16 (𝑧 = 𝑏 → ((𝐵𝑧) ∈ 𝑀 ↔ (𝐵𝑏) ∈ 𝑀))
2524, 5elrab2 3697 . . . . . . . . . . . . . . 15 (𝑏𝐽 ↔ (𝑏 ∈ 𝒫 𝐵 ∧ (𝐵𝑏) ∈ 𝑀))
2625simprbi 496 . . . . . . . . . . . . . 14 (𝑏𝐽 → (𝐵𝑏) ∈ 𝑀)
2726rgen 3060 . . . . . . . . . . . . 13 𝑏𝐽 (𝐵𝑏) ∈ 𝑀
28 ssralv 4063 . . . . . . . . . . . . . 14 (𝑎𝐽 → (∀𝑏𝐽 (𝐵𝑏) ∈ 𝑀 → ∀𝑏𝑎 (𝐵𝑏) ∈ 𝑀))
2928adantl 481 . . . . . . . . . . . . 13 ((𝜑𝑎𝐽) → (∀𝑏𝐽 (𝐵𝑏) ∈ 𝑀 → ∀𝑏𝑎 (𝐵𝑏) ∈ 𝑀))
3027, 29mpi 20 . . . . . . . . . . . 12 ((𝜑𝑎𝐽) → ∀𝑏𝑎 (𝐵𝑏) ∈ 𝑀)
3130adantr 480 . . . . . . . . . . 11 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → ∀𝑏𝑎 (𝐵𝑏) ∈ 𝑀)
32 mreiincl 17640 . . . . . . . . . . 11 ((𝑀 ∈ (Moore‘𝐵) ∧ 𝑎 ≠ ∅ ∧ ∀𝑏𝑎 (𝐵𝑏) ∈ 𝑀) → 𝑏𝑎 (𝐵𝑏) ∈ 𝑀)
3321, 22, 31, 32syl3anc 1370 . . . . . . . . . 10 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → 𝑏𝑎 (𝐵𝑏) ∈ 𝑀)
3419, 33eqeltrrd 2839 . . . . . . . . 9 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → (𝐵 𝑏𝑎 𝑏) ∈ 𝑀)
3517, 34eqeltrid 2842 . . . . . . . 8 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → (𝐵 𝑎) ∈ 𝑀)
36 difeq2 4129 . . . . . . . . . 10 (𝑧 = 𝑎 → (𝐵𝑧) = (𝐵 𝑎))
3736eleq1d 2823 . . . . . . . . 9 (𝑧 = 𝑎 → ((𝐵𝑧) ∈ 𝑀 ↔ (𝐵 𝑎) ∈ 𝑀))
3837, 5elrab2 3697 . . . . . . . 8 ( 𝑎𝐽 ↔ ( 𝑎 ∈ 𝒫 𝐵 ∧ (𝐵 𝑎) ∈ 𝑀))
3915, 35, 38sylanbrc 583 . . . . . . 7 (((𝜑𝑎𝐽) ∧ 𝑎 ≠ ∅) → 𝑎𝐽)
40 0elpw 5361 . . . . . . . . . 10 ∅ ∈ 𝒫 𝐵
4140a1i 11 . . . . . . . . 9 (𝜑 → ∅ ∈ 𝒫 𝐵)
42 mre1cl 17638 . . . . . . . . . 10 (𝑀 ∈ (Moore‘𝐵) → 𝐵𝑀)
4320, 42syl 17 . . . . . . . . 9 (𝜑𝐵𝑀)
44 difeq2 4129 . . . . . . . . . . . 12 (𝑧 = ∅ → (𝐵𝑧) = (𝐵 ∖ ∅))
45 dif0 4383 . . . . . . . . . . . 12 (𝐵 ∖ ∅) = 𝐵
4644, 45eqtrdi 2790 . . . . . . . . . . 11 (𝑧 = ∅ → (𝐵𝑧) = 𝐵)
4746eleq1d 2823 . . . . . . . . . 10 (𝑧 = ∅ → ((𝐵𝑧) ∈ 𝑀𝐵𝑀))
4847, 5elrab2 3697 . . . . . . . . 9 (∅ ∈ 𝐽 ↔ (∅ ∈ 𝒫 𝐵𝐵𝑀))
4941, 43, 48sylanbrc 583 . . . . . . . 8 (𝜑 → ∅ ∈ 𝐽)
5049adantr 480 . . . . . . 7 ((𝜑𝑎𝐽) → ∅ ∈ 𝐽)
514, 39, 50pm2.61ne 3024 . . . . . 6 ((𝜑𝑎𝐽) → 𝑎𝐽)
5251ex 412 . . . . 5 (𝜑 → (𝑎𝐽 𝑎𝐽))
5352alrimiv 1924 . . . 4 (𝜑 → ∀𝑎(𝑎𝐽 𝑎𝐽))
54 inss1 4244 . . . . . . . 8 (𝑎𝑏) ⊆ 𝑎
55 difeq2 4129 . . . . . . . . . . . . 13 (𝑧 = 𝑎 → (𝐵𝑧) = (𝐵𝑎))
5655eleq1d 2823 . . . . . . . . . . . 12 (𝑧 = 𝑎 → ((𝐵𝑧) ∈ 𝑀 ↔ (𝐵𝑎) ∈ 𝑀))
5756, 5elrab2 3697 . . . . . . . . . . 11 (𝑎𝐽 ↔ (𝑎 ∈ 𝒫 𝐵 ∧ (𝐵𝑎) ∈ 𝑀))
5857simplbi 497 . . . . . . . . . 10 (𝑎𝐽𝑎 ∈ 𝒫 𝐵)
5958elpwid 4613 . . . . . . . . 9 (𝑎𝐽𝑎𝐵)
6059ad2antrl 728 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → 𝑎𝐵)
6154, 60sstrid 4006 . . . . . . 7 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → (𝑎𝑏) ⊆ 𝐵)
62 vex 3481 . . . . . . . . 9 𝑎 ∈ V
6362inex1 5322 . . . . . . . 8 (𝑎𝑏) ∈ V
6463elpw 4608 . . . . . . 7 ((𝑎𝑏) ∈ 𝒫 𝐵 ↔ (𝑎𝑏) ⊆ 𝐵)
6561, 64sylibr 234 . . . . . 6 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → (𝑎𝑏) ∈ 𝒫 𝐵)
66 difindi 4297 . . . . . . 7 (𝐵 ∖ (𝑎𝑏)) = ((𝐵𝑎) ∪ (𝐵𝑏))
6757simprbi 496 . . . . . . . . 9 (𝑎𝐽 → (𝐵𝑎) ∈ 𝑀)
6867ad2antrl 728 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → (𝐵𝑎) ∈ 𝑀)
6926ad2antll 729 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → (𝐵𝑏) ∈ 𝑀)
70 simpl 482 . . . . . . . 8 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → 𝜑)
71 uneq1 4170 . . . . . . . . . . . 12 (𝑥 = (𝐵𝑎) → (𝑥𝑦) = ((𝐵𝑎) ∪ 𝑦))
7271eleq1d 2823 . . . . . . . . . . 11 (𝑥 = (𝐵𝑎) → ((𝑥𝑦) ∈ 𝑀 ↔ ((𝐵𝑎) ∪ 𝑦) ∈ 𝑀))
7372imbi2d 340 . . . . . . . . . 10 (𝑥 = (𝐵𝑎) → ((𝜑 → (𝑥𝑦) ∈ 𝑀) ↔ (𝜑 → ((𝐵𝑎) ∪ 𝑦) ∈ 𝑀)))
74 uneq2 4171 . . . . . . . . . . . 12 (𝑦 = (𝐵𝑏) → ((𝐵𝑎) ∪ 𝑦) = ((𝐵𝑎) ∪ (𝐵𝑏)))
7574eleq1d 2823 . . . . . . . . . . 11 (𝑦 = (𝐵𝑏) → (((𝐵𝑎) ∪ 𝑦) ∈ 𝑀 ↔ ((𝐵𝑎) ∪ (𝐵𝑏)) ∈ 𝑀))
7675imbi2d 340 . . . . . . . . . 10 (𝑦 = (𝐵𝑏) → ((𝜑 → ((𝐵𝑎) ∪ 𝑦) ∈ 𝑀) ↔ (𝜑 → ((𝐵𝑎) ∪ (𝐵𝑏)) ∈ 𝑀)))
77 mretopd.u . . . . . . . . . . . 12 ((𝜑𝑥𝑀𝑦𝑀) → (𝑥𝑦) ∈ 𝑀)
78773expb 1119 . . . . . . . . . . 11 ((𝜑 ∧ (𝑥𝑀𝑦𝑀)) → (𝑥𝑦) ∈ 𝑀)
7978expcom 413 . . . . . . . . . 10 ((𝑥𝑀𝑦𝑀) → (𝜑 → (𝑥𝑦) ∈ 𝑀))
8073, 76, 79vtocl2ga 3577 . . . . . . . . 9 (((𝐵𝑎) ∈ 𝑀 ∧ (𝐵𝑏) ∈ 𝑀) → (𝜑 → ((𝐵𝑎) ∪ (𝐵𝑏)) ∈ 𝑀))
8180imp 406 . . . . . . . 8 ((((𝐵𝑎) ∈ 𝑀 ∧ (𝐵𝑏) ∈ 𝑀) ∧ 𝜑) → ((𝐵𝑎) ∪ (𝐵𝑏)) ∈ 𝑀)
8268, 69, 70, 81syl21anc 838 . . . . . . 7 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → ((𝐵𝑎) ∪ (𝐵𝑏)) ∈ 𝑀)
8366, 82eqeltrid 2842 . . . . . 6 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → (𝐵 ∖ (𝑎𝑏)) ∈ 𝑀)
84 difeq2 4129 . . . . . . . 8 (𝑧 = (𝑎𝑏) → (𝐵𝑧) = (𝐵 ∖ (𝑎𝑏)))
8584eleq1d 2823 . . . . . . 7 (𝑧 = (𝑎𝑏) → ((𝐵𝑧) ∈ 𝑀 ↔ (𝐵 ∖ (𝑎𝑏)) ∈ 𝑀))
8685, 5elrab2 3697 . . . . . 6 ((𝑎𝑏) ∈ 𝐽 ↔ ((𝑎𝑏) ∈ 𝒫 𝐵 ∧ (𝐵 ∖ (𝑎𝑏)) ∈ 𝑀))
8765, 83, 86sylanbrc 583 . . . . 5 ((𝜑 ∧ (𝑎𝐽𝑏𝐽)) → (𝑎𝑏) ∈ 𝐽)
8887ralrimivva 3199 . . . 4 (𝜑 → ∀𝑎𝐽𝑏𝐽 (𝑎𝑏) ∈ 𝐽)
8943pwexd 5384 . . . . . 6 (𝜑 → 𝒫 𝐵 ∈ V)
905, 89rabexd 5345 . . . . 5 (𝜑𝐽 ∈ V)
91 istopg 22916 . . . . 5 (𝐽 ∈ V → (𝐽 ∈ Top ↔ (∀𝑎(𝑎𝐽 𝑎𝐽) ∧ ∀𝑎𝐽𝑏𝐽 (𝑎𝑏) ∈ 𝐽)))
9290, 91syl 17 . . . 4 (𝜑 → (𝐽 ∈ Top ↔ (∀𝑎(𝑎𝐽 𝑎𝐽) ∧ ∀𝑎𝐽𝑏𝐽 (𝑎𝑏) ∈ 𝐽)))
9353, 88, 92mpbir2and 713 . . 3 (𝜑𝐽 ∈ Top)
946unissi 4920 . . . . . 6 𝐽 𝒫 𝐵
95 unipw 5460 . . . . . 6 𝒫 𝐵 = 𝐵
9694, 95sseqtri 4031 . . . . 5 𝐽𝐵
97 pwidg 4624 . . . . . . 7 (𝐵𝑀𝐵 ∈ 𝒫 𝐵)
9843, 97syl 17 . . . . . 6 (𝜑𝐵 ∈ 𝒫 𝐵)
99 difid 4381 . . . . . . 7 (𝐵𝐵) = ∅
100 mretopd.z . . . . . . 7 (𝜑 → ∅ ∈ 𝑀)
10199, 100eqeltrid 2842 . . . . . 6 (𝜑 → (𝐵𝐵) ∈ 𝑀)
102 difeq2 4129 . . . . . . . 8 (𝑧 = 𝐵 → (𝐵𝑧) = (𝐵𝐵))
103102eleq1d 2823 . . . . . . 7 (𝑧 = 𝐵 → ((𝐵𝑧) ∈ 𝑀 ↔ (𝐵𝐵) ∈ 𝑀))
104103, 5elrab2 3697 . . . . . 6 (𝐵𝐽 ↔ (𝐵 ∈ 𝒫 𝐵 ∧ (𝐵𝐵) ∈ 𝑀))
10598, 101, 104sylanbrc 583 . . . . 5 (𝜑𝐵𝐽)
106 unissel 4942 . . . . 5 (( 𝐽𝐵𝐵𝐽) → 𝐽 = 𝐵)
10796, 105, 106sylancr 587 . . . 4 (𝜑 𝐽 = 𝐵)
108107eqcomd 2740 . . 3 (𝜑𝐵 = 𝐽)
109 istopon 22933 . . 3 (𝐽 ∈ (TopOn‘𝐵) ↔ (𝐽 ∈ Top ∧ 𝐵 = 𝐽))
11093, 108, 109sylanbrc 583 . 2 (𝜑𝐽 ∈ (TopOn‘𝐵))
111 eqid 2734 . . . . 5 𝐽 = 𝐽
112111cldval 23046 . . . 4 (𝐽 ∈ Top → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝐽 ∣ ( 𝐽𝑥) ∈ 𝐽})
11393, 112syl 17 . . 3 (𝜑 → (Clsd‘𝐽) = {𝑥 ∈ 𝒫 𝐽 ∣ ( 𝐽𝑥) ∈ 𝐽})
114107pweqd 4621 . . . 4 (𝜑 → 𝒫 𝐽 = 𝒫 𝐵)
115107difeq1d 4134 . . . . 5 (𝜑 → ( 𝐽𝑥) = (𝐵𝑥))
116115eleq1d 2823 . . . 4 (𝜑 → (( 𝐽𝑥) ∈ 𝐽 ↔ (𝐵𝑥) ∈ 𝐽))
117114, 116rabeqbidv 3451 . . 3 (𝜑 → {𝑥 ∈ 𝒫 𝐽 ∣ ( 𝐽𝑥) ∈ 𝐽} = {𝑥 ∈ 𝒫 𝐵 ∣ (𝐵𝑥) ∈ 𝐽})
1185eleq2i 2830 . . . . . . 7 ((𝐵𝑥) ∈ 𝐽 ↔ (𝐵𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ 𝑀})
119 difss 4145 . . . . . . . . . 10 (𝐵𝑥) ⊆ 𝐵
120 elpw2g 5338 . . . . . . . . . . 11 (𝐵𝑀 → ((𝐵𝑥) ∈ 𝒫 𝐵 ↔ (𝐵𝑥) ⊆ 𝐵))
12143, 120syl 17 . . . . . . . . . 10 (𝜑 → ((𝐵𝑥) ∈ 𝒫 𝐵 ↔ (𝐵𝑥) ⊆ 𝐵))
122119, 121mpbiri 258 . . . . . . . . 9 (𝜑 → (𝐵𝑥) ∈ 𝒫 𝐵)
123 difeq2 4129 . . . . . . . . . . 11 (𝑧 = (𝐵𝑥) → (𝐵𝑧) = (𝐵 ∖ (𝐵𝑥)))
124123eleq1d 2823 . . . . . . . . . 10 (𝑧 = (𝐵𝑥) → ((𝐵𝑧) ∈ 𝑀 ↔ (𝐵 ∖ (𝐵𝑥)) ∈ 𝑀))
125124elrab3 3695 . . . . . . . . 9 ((𝐵𝑥) ∈ 𝒫 𝐵 → ((𝐵𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ 𝑀} ↔ (𝐵 ∖ (𝐵𝑥)) ∈ 𝑀))
126122, 125syl 17 . . . . . . . 8 (𝜑 → ((𝐵𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ 𝑀} ↔ (𝐵 ∖ (𝐵𝑥)) ∈ 𝑀))
127126adantr 480 . . . . . . 7 ((𝜑𝑥 ∈ 𝒫 𝐵) → ((𝐵𝑥) ∈ {𝑧 ∈ 𝒫 𝐵 ∣ (𝐵𝑧) ∈ 𝑀} ↔ (𝐵 ∖ (𝐵𝑥)) ∈ 𝑀))
128118, 127bitrid 283 . . . . . 6 ((𝜑𝑥 ∈ 𝒫 𝐵) → ((𝐵𝑥) ∈ 𝐽 ↔ (𝐵 ∖ (𝐵𝑥)) ∈ 𝑀))
129 elpwi 4611 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝐵𝑥𝐵)
130 dfss4 4274 . . . . . . . . 9 (𝑥𝐵 ↔ (𝐵 ∖ (𝐵𝑥)) = 𝑥)
131129, 130sylib 218 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐵 → (𝐵 ∖ (𝐵𝑥)) = 𝑥)
132131adantl 481 . . . . . . 7 ((𝜑𝑥 ∈ 𝒫 𝐵) → (𝐵 ∖ (𝐵𝑥)) = 𝑥)
133132eleq1d 2823 . . . . . 6 ((𝜑𝑥 ∈ 𝒫 𝐵) → ((𝐵 ∖ (𝐵𝑥)) ∈ 𝑀𝑥𝑀))
134128, 133bitrd 279 . . . . 5 ((𝜑𝑥 ∈ 𝒫 𝐵) → ((𝐵𝑥) ∈ 𝐽𝑥𝑀))
135134rabbidva 3439 . . . 4 (𝜑 → {𝑥 ∈ 𝒫 𝐵 ∣ (𝐵𝑥) ∈ 𝐽} = {𝑥 ∈ 𝒫 𝐵𝑥𝑀})
136 incom 4216 . . . . . 6 (𝑀 ∩ 𝒫 𝐵) = (𝒫 𝐵𝑀)
137 dfin5 3970 . . . . . 6 (𝒫 𝐵𝑀) = {𝑥 ∈ 𝒫 𝐵𝑥𝑀}
138136, 137eqtri 2762 . . . . 5 (𝑀 ∩ 𝒫 𝐵) = {𝑥 ∈ 𝒫 𝐵𝑥𝑀}
139 mresspw 17636 . . . . . . 7 (𝑀 ∈ (Moore‘𝐵) → 𝑀 ⊆ 𝒫 𝐵)
14020, 139syl 17 . . . . . 6 (𝜑𝑀 ⊆ 𝒫 𝐵)
141 dfss2 3980 . . . . . 6 (𝑀 ⊆ 𝒫 𝐵 ↔ (𝑀 ∩ 𝒫 𝐵) = 𝑀)
142140, 141sylib 218 . . . . 5 (𝜑 → (𝑀 ∩ 𝒫 𝐵) = 𝑀)
143138, 142eqtr3id 2788 . . . 4 (𝜑 → {𝑥 ∈ 𝒫 𝐵𝑥𝑀} = 𝑀)
144135, 143eqtrd 2774 . . 3 (𝜑 → {𝑥 ∈ 𝒫 𝐵 ∣ (𝐵𝑥) ∈ 𝐽} = 𝑀)
145113, 117, 1443eqtrrd 2779 . 2 (𝜑𝑀 = (Clsd‘𝐽))
146110, 145jca 511 1 (𝜑 → (𝐽 ∈ (TopOn‘𝐵) ∧ 𝑀 = (Clsd‘𝐽)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086  wal 1534   = wceq 1536  wcel 2105  wne 2937  wral 3058  {crab 3432  Vcvv 3477  cdif 3959  cun 3960  cin 3961  wss 3962  c0 4338  𝒫 cpw 4604   cuni 4911   ciun 4995   ciin 4996  cfv 6562  Moorecmre 17626  Topctop 22914  TopOnctopon 22931  Clsdccld 23039
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-iin 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-iota 6515  df-fun 6564  df-fv 6570  df-mre 17630  df-top 22915  df-topon 22932  df-cld 23042
This theorem is referenced by:  iscldtop  23118  zartopn  33835  istopclsd  42687
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