Proof of Theorem fclsopn
Step | Hyp | Ref
| Expression |
1 | | isfcls2 23164 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))) |
2 | | filn0 23013 |
. . . . . 6
⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ≠ ∅) |
3 | 2 | adantl 482 |
. . . . 5
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → 𝐹 ≠ ∅) |
4 | | r19.2z 4425 |
. . . . . 6
⊢ ((𝐹 ≠ ∅ ∧
∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) |
5 | 4 | ex 413 |
. . . . 5
⊢ (𝐹 ≠ ∅ →
(∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))) |
6 | 3, 5 | syl 17 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → ∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠))) |
7 | | topontop 22062 |
. . . . . . . . 9
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) |
8 | 7 | ad2antrr 723 |
. . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → 𝐽 ∈ Top) |
9 | | filelss 23003 |
. . . . . . . . . 10
⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ 𝑋) |
10 | 9 | adantll 711 |
. . . . . . . . 9
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ 𝑋) |
11 | | toponuni 22063 |
. . . . . . . . . 10
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) |
12 | 11 | ad2antrr 723 |
. . . . . . . . 9
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → 𝑋 = ∪ 𝐽) |
13 | 10, 12 | sseqtrd 3961 |
. . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ ∪ 𝐽) |
14 | | eqid 2738 |
. . . . . . . . 9
⊢ ∪ 𝐽 =
∪ 𝐽 |
15 | 14 | clsss3 22210 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ ∪ 𝐽)
→ ((cls‘𝐽)‘𝑠) ⊆ ∪ 𝐽) |
16 | 8, 13, 15 | syl2anc 584 |
. . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → ((cls‘𝐽)‘𝑠) ⊆ ∪ 𝐽) |
17 | 16, 12 | sseqtrrd 3962 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → ((cls‘𝐽)‘𝑠) ⊆ 𝑋) |
18 | 17 | sseld 3920 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝑠 ∈ 𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ 𝑋)) |
19 | 18 | rexlimdva 3213 |
. . . 4
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∃𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ 𝑋)) |
20 | 6, 19 | syld 47 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) → 𝐴 ∈ 𝑋)) |
21 | 20 | pm4.71rd 563 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)))) |
22 | 7 | ad3antrrr 727 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝐽 ∈ Top) |
23 | 13 | adantlr 712 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑠 ⊆ ∪ 𝐽) |
24 | | simplr 766 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝐴 ∈ 𝑋) |
25 | 11 | ad3antrrr 727 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝑋 = ∪ 𝐽) |
26 | 24, 25 | eleqtrd 2841 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → 𝐴 ∈ ∪ 𝐽) |
27 | 14 | elcls 22224 |
. . . . . 6
⊢ ((𝐽 ∈ Top ∧ 𝑠 ⊆ ∪ 𝐽
∧ 𝐴 ∈ ∪ 𝐽)
→ (𝐴 ∈
((cls‘𝐽)‘𝑠) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅))) |
28 | 22, 23, 26, 27 | syl3anc 1370 |
. . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) ∧ 𝑠 ∈ 𝐹) → (𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅))) |
29 | 28 | ralbidva 3111 |
. . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑠 ∈ 𝐹 ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅))) |
30 | | ralcom 3166 |
. . . . 5
⊢
(∀𝑠 ∈
𝐹 ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅) ↔ ∀𝑜 ∈ 𝐽 ∀𝑠 ∈ 𝐹 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅)) |
31 | | r19.21v 3113 |
. . . . . 6
⊢
(∀𝑠 ∈
𝐹 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅) ↔ (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)) |
32 | 31 | ralbii 3092 |
. . . . 5
⊢
(∀𝑜 ∈
𝐽 ∀𝑠 ∈ 𝐹 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)) |
33 | 30, 32 | bitri 274 |
. . . 4
⊢
(∀𝑠 ∈
𝐹 ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → (𝑜 ∩ 𝑠) ≠ ∅) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)) |
34 | 29, 33 | bitrdi 287 |
. . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐴 ∈ 𝑋) → (∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠) ↔ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅))) |
35 | 34 | pm5.32da 579 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝐴 ∈ 𝑋 ∧ ∀𝑠 ∈ 𝐹 𝐴 ∈ ((cls‘𝐽)‘𝑠)) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))) |
36 | 1, 21, 35 | 3bitrd 305 |
1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐴 ∈ (𝐽 fClus 𝐹) ↔ (𝐴 ∈ 𝑋 ∧ ∀𝑜 ∈ 𝐽 (𝐴 ∈ 𝑜 → ∀𝑠 ∈ 𝐹 (𝑜 ∩ 𝑠) ≠ ∅)))) |