| Step | Hyp | Ref
| Expression |
| 1 | | eqid 2737 |
. . . . 5
⊢ ∪ 𝐽 =
∪ 𝐽 |
| 2 | 1 | fclscmpi 24037 |
. . . 4
⊢ ((𝐽 ∈ Comp ∧ 𝑓 ∈ (Fil‘∪ 𝐽))
→ (𝐽 fClus 𝑓) ≠ ∅) |
| 3 | 2 | ralrimiva 3146 |
. . 3
⊢ (𝐽 ∈ Comp →
∀𝑓 ∈
(Fil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅) |
| 4 | | toponuni 22920 |
. . . . 5
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) |
| 5 | 4 | fveq2d 6910 |
. . . 4
⊢ (𝐽 ∈ (TopOn‘𝑋) → (Fil‘𝑋) = (Fil‘∪ 𝐽)) |
| 6 | 5 | raleqdv 3326 |
. . 3
⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ ↔ ∀𝑓 ∈ (Fil‘∪ 𝐽)(𝐽 fClus 𝑓) ≠ ∅)) |
| 7 | 3, 6 | imbitrrid 246 |
. 2
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp → ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅)) |
| 8 | | elpwi 4607 |
. . . . . 6
⊢ (𝑥 ∈ 𝒫
(Clsd‘𝐽) → 𝑥 ⊆ (Clsd‘𝐽)) |
| 9 | | vn0 4345 |
. . . . . . . . . 10
⊢ V ≠
∅ |
| 10 | | simpr 484 |
. . . . . . . . . . . . 13
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → 𝑥 = ∅) |
| 11 | 10 | inteqd 4951 |
. . . . . . . . . . . 12
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → ∩ 𝑥 =
∩ ∅) |
| 12 | | int0 4962 |
. . . . . . . . . . . 12
⊢ ∩ ∅ = V |
| 13 | 11, 12 | eqtrdi 2793 |
. . . . . . . . . . 11
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → ∩ 𝑥 =
V) |
| 14 | 13 | neeq1d 3000 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 = ∅) → (∩ 𝑥
≠ ∅ ↔ V ≠ ∅)) |
| 15 | 9, 14 | mpbiri 258 |
. . . . . . . . 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 9459 |
. . . . . . . . . . . . . . . 16
⊢ (𝑥 ∈ V → 𝑥 ⊆ (fi‘𝑥)) |
| 18 | 17 | elv 3485 |
. . . . . . . . . . . . . . 15
⊢ 𝑥 ⊆ (fi‘𝑥) |
| 19 | | simplrl 777 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ (Clsd‘𝐽)) |
| 20 | 1 | cldss2 23038 |
. . . . . . . . . . . . . . . . . . 19
⊢
(Clsd‘𝐽)
⊆ 𝒫 ∪ 𝐽 |
| 21 | 4 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑋 = ∪ 𝐽) |
| 22 | 21 | pweqd 4617 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝒫 𝑋 = 𝒫 ∪ 𝐽) |
| 23 | 20, 22 | sseqtrrid 4027 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (Clsd‘𝐽) ⊆ 𝒫 𝑋) |
| 24 | 19, 23 | sstrd 3994 |
. . . . . . . . . . . . . . . . 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 778 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ¬ ∅ ∈
(fi‘𝑥)) |
| 27 | | toponmax 22932 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) |
| 28 | 27 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑋 ∈ 𝐽) |
| 29 | | fsubbas 23875 |
. . . . . . . . . . . . . . . . . 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 1343 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (fi‘𝑥) ∈ (fBas‘𝑋)) |
| 32 | | ssfg 23880 |
. . . . . . . . . . . . . . . 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 3995 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → 𝑥 ⊆ (𝑋filGen(fi‘𝑥))) |
| 35 | 34 | sselda 3983 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (𝑋filGen(fi‘𝑥))) |
| 36 | | fclssscls 24026 |
. . . . . . . . . . . . 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 3983 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ (Clsd‘𝐽)) |
| 39 | | cldcls 23050 |
. . . . . . . . . . . . 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 4020 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) ∧ 𝑦 ∈ 𝑥) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦) |
| 42 | 41 | ralrimiva 3146 |
. . . . . . . . . 10
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → ∀𝑦 ∈ 𝑥 (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦) |
| 43 | | ssint 4964 |
. . . . . . . . . 10
⊢ ((𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ∩
𝑥 ↔ ∀𝑦 ∈ 𝑥 (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ 𝑦) |
| 44 | 42, 43 | sylibr 234 |
. . . . . . . . 9
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ∩
𝑥) |
| 45 | | fgcl 23886 |
. . . . . . . . . 10
⊢
((fi‘𝑥) ∈
(fBas‘𝑋) →
(𝑋filGen(fi‘𝑥)) ∈ (Fil‘𝑋)) |
| 46 | | oveq2 7439 |
. . . . . . . . . . . 12
⊢ (𝑓 = (𝑋filGen(fi‘𝑥)) → (𝐽 fClus 𝑓) = (𝐽 fClus (𝑋filGen(fi‘𝑥)))) |
| 47 | 46 | neeq1d 3000 |
. . . . . . . . . . 11
⊢ (𝑓 = (𝑋filGen(fi‘𝑥)) → ((𝐽 fClus 𝑓) ≠ ∅ ↔ (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅)) |
| 48 | 47 | rspcv 3618 |
. . . . . . . . . 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 4404 |
. . . . . . . . 9
⊢ (((𝐽 fClus (𝑋filGen(fi‘𝑥))) ⊆ ∩
𝑥 ∧ (𝐽 fClus (𝑋filGen(fi‘𝑥))) ≠ ∅) → ∩ 𝑥
≠ ∅) |
| 51 | 44, 49, 50 | syl6an 684 |
. . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑥 ⊆ (Clsd‘𝐽) ∧ ¬ ∅ ∈ (fi‘𝑥))) ∧ 𝑥 ≠ ∅) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∩ 𝑥
≠ ∅)) |
| 52 | 16, 51 | pm2.61dane 3029 |
. . . . . . 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 593 |
. . . . 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 3154 |
. . 3
⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → ∀𝑥 ∈ 𝒫
(Clsd‘𝐽)(¬
∅ ∈ (fi‘𝑥)
→ ∩ 𝑥 ≠ ∅))) |
| 57 | | topontop 22919 |
. . . 4
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) |
| 58 | | cmpfi 23416 |
. . . 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 259 |
. 2
⊢ (𝐽 ∈ (TopOn‘𝑋) → (∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅ → 𝐽 ∈ Comp)) |
| 61 | 7, 60 | impbid 212 |
1
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Comp ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐽 fClus 𝑓) ≠ ∅)) |