Step | Hyp | Ref
| Expression |
1 | | eqid 2739 |
. . . . 5
⊢ ∪ 𝐽 =
∪ 𝐽 |
2 | 1 | fclscmpi 23161 |
. . . 4
⊢ ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Fil‘∪ 𝐽))
→ (𝐽 fClus 𝑓) ≠ ∅) |
3 | 2 | ralrimiva 3109 |
. . 3
⊢ (𝐽 ∈ Comp →
∀𝑓 ∈
(Fil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅) |
4 | | toponuni 22044 |
. . . . 5
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) |
5 | 4 | fveq2d 6772 |
. . . 4
⊢ (𝐽 ∈ (TopOn‘𝑋) → (Fil‘𝑋) = (Fil‘∪ 𝐽)) |
6 | 5 | raleqdv 3346 |
. . 3
⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (Fil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)) |
7 | 3, 6 | syl5ibr 245 |
. 2
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp → ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅)) |
8 | | elpwi 4547 |
. . . . . 6
⊢ (𝑥 ∈ 𝒫
(Clsd‘𝐽) → 𝑥 ⊆ (Clsd‘𝐽)) |
9 | | vn0 4277 |
. . . . . . . . . 10
⊢ V ≠
∅ |
10 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → 𝑥 = ∅) |
11 | 10 | inteqd 4889 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → ∩ 𝑥 =
∩ ∅) |
12 | | int0 4898 |
. . . . . . . . . . . 12
⊢ ∩ ∅ = V |
13 | 11, 12 | eqtrdi 2795 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → ∩ 𝑥 =
V) |
14 | 13 | neeq1d 3004 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → (∩ 𝑥
≠ ∅ ↔ V ≠ ∅)) |
15 | 9, 14 | mpbiri 257 |
. . . . . . . . 9
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → ∩ 𝑥
≠ ∅) |
16 | 15 | a1d 25 |
. . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥
≠ ∅)) |
17 | | ssfii 9139 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ V → 𝑥 ⊆ (fi‘𝑥)) |
18 | 17 | elv 3436 |
. . . . . . . . . . . . . . 15
⊢ 𝑥 ⊆ (fi‘𝑥) |
19 | | simplrl 773 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ (Clsd‘𝐽)) |
20 | 1 | cldss2 22162 |
. . . . . . . . . . . . . . . . . . 19
⊢
(Clsd‘𝐽)
⊆ 𝒫 ∪ 𝐽 |
21 | 4 | ad2antrr 722 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑋 = ∪ 𝐽) |
22 | 21 | pweqd 4557 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝒫 𝑋 = 𝒫 ∪ 𝐽) |
23 | 20, 22 | sseqtrrid 3978 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (Clsd‘𝐽) ⊆ 𝒫 𝑋) |
24 | 19, 23 | sstrd 3935 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ 𝒫 𝑋) |
25 | | simpr 484 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ≠ ∅) |
26 | | simplrr 774 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ¬ ∅ ∈
(fi‘𝑥)) |
27 | | toponmax 22056 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) |
28 | 27 | ad2antrr 722 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑋 ∈ 𝐽) |
29 | | fsubbas 22999 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑋 ∈ 𝐽 → ((fi‘𝑥) ∈ (fBas‘𝑋) ↔ (𝑥 ⊆ 𝒫 𝑋 ∧ 𝑥 ≠ ∅ ∧ ¬ ∅ ∈
(fi‘𝑥)))) |
30 | 28, 29 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ((fi‘𝑥) ∈ (fBas‘𝑋) ↔ (𝑥 ⊆ 𝒫 𝑋 ∧ 𝑥 ≠ ∅ ∧ ¬ ∅ ∈
(fi‘𝑥)))) |
31 | 24, 25, 26, 30 | mpbir3and 1340 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (fi‘𝑥) ∈ (fBas‘𝑋)) |
32 | | ssfg 23004 |
. . . . . . . . . . . . . . . 16
⊢
((fi‘𝑥) ∈
(fBas‘𝑋) →
(fi‘𝑥) ⊆ (𝑋filGen(fi‘𝑥))) |
33 | 31, 32 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (fi‘𝑥) ⊆ (𝑋filGen(fi‘𝑥))) |
34 | 18, 33 | sstrid 3936 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ (𝑋filGen(fi‘𝑥))) |
35 | 34 | sselda 3925 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝑋filGen(fi‘𝑥))) |
36 | | fclssscls 23150 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (𝑋filGen(fi‘𝑥)) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ((cls‘𝐽)‘𝑦)) |
37 | 35, 36 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ((cls‘𝐽)‘𝑦)) |
38 | 19 | sselda 3925 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (Clsd‘𝐽)) |
39 | | cldcls 22174 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑦) = 𝑦) |
40 | 38, 39 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → ((cls‘𝐽)‘𝑦) = 𝑦) |
41 | 37, 40 | sseqtrd 3965 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦) |
42 | 41 | ralrimiva 3109 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ∀𝑦 ∈ 𝑥 (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦) |
43 | | ssint 4900 |
. . . . . . . . . 10
⊢ ((𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ∩
𝑥 ↔ ∀𝑦 ∈ 𝑥 (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦) |
44 | 42, 43 | sylibr 233 |
. . . . . . . . 9
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ∩
𝑥) |
45 | | fgcl 23010 |
. . . . . . . . . 10
⊢
((fi‘𝑥) ∈
(fBas‘𝑋) →
(𝑋filGen(fi‘𝑥)) ∈ (Fil‘𝑋)) |
46 | | oveq2 7276 |
. . . . . . . . . . . 12
⊢ (𝑓 = (𝑋filGen(fi‘𝑥)) → (𝐽 fClus 𝑓) = (𝐽 fClus (𝑋filGen(fi‘𝑥)))) |
47 | 46 | neeq1d 3004 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝑋filGen(fi‘𝑥)) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅)) |
48 | 47 | rspcv 3555 |
. . . . . . . . . 10
⊢ ((𝑋filGen(fi‘𝑥)) ∈ (Fil‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅)) |
49 | 31, 45, 48 | 3syl 18 |
. . . . . . . . 9
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅)) |
50 | | ssn0 4339 |
. . . . . . . . 9
⊢ (((𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ∩
𝑥 ∧ (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅) → ∩ 𝑥
≠ ∅) |
51 | 44, 49, 50 | syl6an 680 |
. . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥
≠ ∅)) |
52 | 16, 51 | pm2.61dane 3033 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥
≠ ∅)) |
53 | 52 | expr 456 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ⊆ (Clsd‘𝐽)) → (¬ ∅ ∈
(fi‘𝑥) →
(∀𝑓 ∈
(Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥
≠ ∅))) |
54 | 8, 53 | sylan2 592 |
. . . . 5
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (¬ ∅ ∈
(fi‘𝑥) →
(∀𝑓 ∈
(Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥
≠ ∅))) |
55 | 54 | com23 86 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝒫 (Clsd‘𝐽)) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → (¬ ∅ ∈
(fi‘𝑥) → ∩ 𝑥
≠ ∅))) |
56 | 55 | ralrimdva 3114 |
. . 3
⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑥 ∈ 𝒫
(Clsd‘𝐽)(¬
∅ ∈ (fi‘𝑥)
→ ∩ 𝑥 ≠ ∅))) |
57 | | topontop 22043 |
. . . 4
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) |
58 | | cmpfi 22540 |
. . . 4
⊢ (𝐽 ∈ Top → (𝐽 ∈ Comp ↔
∀𝑥 ∈ 𝒫
(Clsd‘𝐽)(¬
∅ ∈ (fi‘𝑥)
→ ∩ 𝑥 ≠ ∅))) |
59 | 57, 58 | syl 17 |
. . 3
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑥 ∈ 𝒫
(Clsd‘𝐽)(¬
∅ ∈ (fi‘𝑥)
→ ∩ 𝑥 ≠ ∅))) |
60 | 56, 59 | sylibrd 258 |
. 2
⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝐽 ∈ Comp)) |
61 | 7, 60 | impbid 211 |
1
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅)) |