Step | Hyp | Ref
| Expression |
1 | | simpll 757 |
. . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐽 ∈ (TopOn‘𝑋)) |
2 | | simplr 759 |
. . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐾 ∈ (TopOn‘𝑋)) |
3 | | fclstopon 22224 |
. . . . . . . . 9
⊢ (𝑥 ∈ (𝐾 fClus 𝑓) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝑓 ∈ (Fil‘𝑋))) |
4 | 3 | ad2antll 719 |
. . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → (𝐾 ∈ (TopOn‘𝑋) ↔ 𝑓 ∈ (Fil‘𝑋))) |
5 | 2, 4 | mpbid 224 |
. . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑓 ∈ (Fil‘𝑋)) |
6 | | simprl 761 |
. . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → 𝐽 ⊆ 𝐾) |
7 | | fclsss1 22234 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) |
8 | 1, 5, 6, 7 | syl3anc 1439 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) |
9 | | simprr 763 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑥 ∈ (𝐾 fClus 𝑓)) |
10 | 8, 9 | sseldd 3822 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ (𝐽 ⊆ 𝐾 ∧ 𝑥 ∈ (𝐾 fClus 𝑓))) → 𝑥 ∈ (𝐽 fClus 𝑓)) |
11 | 10 | expr 450 |
. . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) → (𝑥 ∈ (𝐾 fClus 𝑓) → 𝑥 ∈ (𝐽 fClus 𝑓))) |
12 | 11 | ssrdv 3827 |
. . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) |
13 | 12 | ralrimivw 3149 |
. 2
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) |
14 | | simpllr 766 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑋)) |
15 | | toponmax 21138 |
. . . . . . . . 9
⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐾) |
16 | | ssid 3842 |
. . . . . . . . . . 11
⊢ 𝑋 ⊆ 𝑋 |
17 | | eleq2 2848 |
. . . . . . . . . . . . 13
⊢ (𝑢 = 𝑋 → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ 𝑋)) |
18 | | sseq1 3845 |
. . . . . . . . . . . . 13
⊢ (𝑢 = 𝑋 → (𝑢 ⊆ 𝑋 ↔ 𝑋 ⊆ 𝑋)) |
19 | 17, 18 | anbi12d 624 |
. . . . . . . . . . . 12
⊢ (𝑢 = 𝑋 → ((𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋) ↔ (𝑦 ∈ 𝑋 ∧ 𝑋 ⊆ 𝑋))) |
20 | 19 | rspcev 3511 |
. . . . . . . . . . 11
⊢ ((𝑋 ∈ 𝐾 ∧ (𝑦 ∈ 𝑋 ∧ 𝑋 ⊆ 𝑋)) → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋)) |
21 | 16, 20 | mpanr2 694 |
. . . . . . . . . 10
⊢ ((𝑋 ∈ 𝐾 ∧ 𝑦 ∈ 𝑋) → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋)) |
22 | 21 | ex 403 |
. . . . . . . . 9
⊢ (𝑋 ∈ 𝐾 → (𝑦 ∈ 𝑋 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋))) |
23 | 14, 15, 22 | 3syl 18 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → (𝑦 ∈ 𝑋 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋))) |
24 | | eleq2 2848 |
. . . . . . . . 9
⊢ (𝑥 = 𝑋 → (𝑦 ∈ 𝑥 ↔ 𝑦 ∈ 𝑋)) |
25 | | sseq2 3846 |
. . . . . . . . . . 11
⊢ (𝑥 = 𝑋 → (𝑢 ⊆ 𝑥 ↔ 𝑢 ⊆ 𝑋)) |
26 | 25 | anbi2d 622 |
. . . . . . . . . 10
⊢ (𝑥 = 𝑋 → ((𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥) ↔ (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋))) |
27 | 26 | rexbidv 3237 |
. . . . . . . . 9
⊢ (𝑥 = 𝑋 → (∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥) ↔ ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋))) |
28 | 24, 27 | imbi12d 336 |
. . . . . . . 8
⊢ (𝑥 = 𝑋 → ((𝑦 ∈ 𝑥 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥)) ↔ (𝑦 ∈ 𝑋 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑋)))) |
29 | 23, 28 | syl5ibrcom 239 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → (𝑥 = 𝑋 → (𝑦 ∈ 𝑥 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥)))) |
30 | | simplll 765 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝐽 ∈ (TopOn‘𝑋)) |
31 | | simprl 761 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝑥 ∈ 𝐽) |
32 | | simprrr 772 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝑦 ∈ 𝑥) |
33 | | supnfcls 22232 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → ¬ 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦})) |
34 | 30, 31, 32, 33 | syl3anc 1439 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → ¬ 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦})) |
35 | | simpllr 766 |
. . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝐾 ∈ (TopOn‘𝑋)) |
36 | | toponmax 21138 |
. . . . . . . . . . . . . . . 16
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 ∈ 𝐽) |
37 | 30, 36 | syl 17 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝑋 ∈ 𝐽) |
38 | | difssd 3961 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (𝑋 ∖ 𝑥) ⊆ 𝑋) |
39 | | toponss 21139 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋) |
40 | 30, 31, 39 | syl2anc 579 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝑥 ⊆ 𝑋) |
41 | | simprrl 771 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝑥 ≠ 𝑋) |
42 | | pssdifn0 4174 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑥 ⊆ 𝑋 ∧ 𝑥 ≠ 𝑋) → (𝑋 ∖ 𝑥) ≠ ∅) |
43 | 40, 41, 42 | syl2anc 579 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (𝑋 ∖ 𝑥) ≠ ∅) |
44 | | supfil 22107 |
. . . . . . . . . . . . . . 15
⊢ ((𝑋 ∈ 𝐽 ∧ (𝑋 ∖ 𝑥) ⊆ 𝑋 ∧ (𝑋 ∖ 𝑥) ≠ ∅) → {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} ∈ (Fil‘𝑋)) |
45 | 37, 38, 43, 44 | syl3anc 1439 |
. . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} ∈ (Fil‘𝑋)) |
46 | | fclsopn 22226 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} ∈ (Fil‘𝑋)) → (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}) ↔ (𝑦 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅)))) |
47 | 35, 45, 46 | syl2anc 579 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}) ↔ (𝑦 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅)))) |
48 | 40, 32 | sseldd 3822 |
. . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → 𝑦 ∈ 𝑋) |
49 | 48 | biantrurd 528 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅) ↔ (𝑦 ∈ 𝑋 ∧ ∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅)))) |
50 | 47, 49 | bitr4d 274 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}) ↔ ∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅))) |
51 | | oveq2 6930 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} → (𝐾 fClus 𝑓) = (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦})) |
52 | | oveq2 6930 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} → (𝐽 fClus 𝑓) = (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦})) |
53 | 51, 52 | sseq12d 3853 |
. . . . . . . . . . . . . 14
⊢ (𝑓 = {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} → ((𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓) ↔ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}) ⊆ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}))) |
54 | | simplr 759 |
. . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) |
55 | 53, 54, 45 | rspcdva 3517 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}) ⊆ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦})) |
56 | 55 | sseld 3820 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (𝑦 ∈ (𝐾 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}) → 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}))) |
57 | 50, 56 | sylbird 252 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅) → 𝑦 ∈ (𝐽 fClus {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦}))) |
58 | 34, 57 | mtod 190 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → ¬ ∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅)) |
59 | | rexanali 3179 |
. . . . . . . . . . 11
⊢
(∃𝑢 ∈
𝐾 (𝑦 ∈ 𝑢 ∧ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅) ↔ ¬ ∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅)) |
60 | | rexnal 3176 |
. . . . . . . . . . . . . 14
⊢
(∃𝑛 ∈
{𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} ¬ (𝑢 ∩ 𝑛) ≠ ∅ ↔ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅) |
61 | | sseq2 3846 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑦 = 𝑛 → ((𝑋 ∖ 𝑥) ⊆ 𝑦 ↔ (𝑋 ∖ 𝑥) ⊆ 𝑛)) |
62 | 61 | elrab 3572 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} ↔ (𝑛 ∈ 𝒫 𝑋 ∧ (𝑋 ∖ 𝑥) ⊆ 𝑛)) |
63 | 62 | simprbi 492 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} → (𝑋 ∖ 𝑥) ⊆ 𝑛) |
64 | | sslin 4059 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑋 ∖ 𝑥) ⊆ 𝑛 → (𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛)) |
65 | 63, 64 | syl 17 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} → (𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛)) |
66 | | ssn0 4202 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛) ∧ (𝑢 ∩ (𝑋 ∖ 𝑥)) ≠ ∅) → (𝑢 ∩ 𝑛) ≠ ∅) |
67 | 66 | ex 403 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛) → ((𝑢 ∩ (𝑋 ∖ 𝑥)) ≠ ∅ → (𝑢 ∩ 𝑛) ≠ ∅)) |
68 | 67 | necon1bd 2987 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛) → (¬ (𝑢 ∩ 𝑛) ≠ ∅ → (𝑢 ∩ (𝑋 ∖ 𝑥)) = ∅)) |
69 | | inssdif0 4178 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑢 ∩ 𝑋) ⊆ 𝑥 ↔ (𝑢 ∩ (𝑋 ∖ 𝑥)) = ∅) |
70 | 68, 69 | syl6ibr 244 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛) → (¬ (𝑢 ∩ 𝑛) ≠ ∅ → (𝑢 ∩ 𝑋) ⊆ 𝑥)) |
71 | | toponss 21139 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑢 ∈ 𝐾) → 𝑢 ⊆ 𝑋) |
72 | 35, 71 | sylan 575 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → 𝑢 ⊆ 𝑋) |
73 | | df-ss 3806 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑢 ⊆ 𝑋 ↔ (𝑢 ∩ 𝑋) = 𝑢) |
74 | 72, 73 | sylib 210 |
. . . . . . . . . . . . . . . . . . 19
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → (𝑢 ∩ 𝑋) = 𝑢) |
75 | 74 | sseq1d 3851 |
. . . . . . . . . . . . . . . . . 18
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → ((𝑢 ∩ 𝑋) ⊆ 𝑥 ↔ 𝑢 ⊆ 𝑥)) |
76 | 75 | biimpd 221 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → ((𝑢 ∩ 𝑋) ⊆ 𝑥 → 𝑢 ⊆ 𝑥)) |
77 | 70, 76 | syl9r 78 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → ((𝑢 ∩ (𝑋 ∖ 𝑥)) ⊆ (𝑢 ∩ 𝑛) → (¬ (𝑢 ∩ 𝑛) ≠ ∅ → 𝑢 ⊆ 𝑥))) |
78 | 65, 77 | syl5 34 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → (𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} → (¬ (𝑢 ∩ 𝑛) ≠ ∅ → 𝑢 ⊆ 𝑥))) |
79 | 78 | rexlimdv 3212 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → (∃𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} ¬ (𝑢 ∩ 𝑛) ≠ ∅ → 𝑢 ⊆ 𝑥)) |
80 | 60, 79 | syl5bir 235 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → (¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅ → 𝑢 ⊆ 𝑥)) |
81 | 80 | anim2d 605 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) ∧ 𝑢 ∈ 𝐾) → ((𝑦 ∈ 𝑢 ∧ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅) → (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥))) |
82 | 81 | reximdva 3198 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ ¬ ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅) → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥))) |
83 | 59, 82 | syl5bir 235 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → (¬ ∀𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 → ∀𝑛 ∈ {𝑦 ∈ 𝒫 𝑋 ∣ (𝑋 ∖ 𝑥) ⊆ 𝑦} (𝑢 ∩ 𝑛) ≠ ∅) → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥))) |
84 | 58, 83 | mpd 15 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥))) → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥)) |
85 | 84 | anassrs 461 |
. . . . . . . 8
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) ∧ (𝑥 ≠ 𝑋 ∧ 𝑦 ∈ 𝑥)) → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥)) |
86 | 85 | exp32 413 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → (𝑥 ≠ 𝑋 → (𝑦 ∈ 𝑥 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥)))) |
87 | 29, 86 | pm2.61dne 3056 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → (𝑦 ∈ 𝑥 → ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥))) |
88 | 87 | ralrimiv 3147 |
. . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥)) |
89 | | topontop 21125 |
. . . . . 6
⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top) |
90 | | eltop2 21187 |
. . . . . 6
⊢ (𝐾 ∈ Top → (𝑥 ∈ 𝐾 ↔ ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥))) |
91 | 14, 89, 90 | 3syl 18 |
. . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∈ 𝐾 ↔ ∀𝑦 ∈ 𝑥 ∃𝑢 ∈ 𝐾 (𝑦 ∈ 𝑢 ∧ 𝑢 ⊆ 𝑥))) |
92 | 88, 91 | mpbird 249 |
. . . 4
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐾) |
93 | 92 | ex 403 |
. . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) → (𝑥 ∈ 𝐽 → 𝑥 ∈ 𝐾)) |
94 | 93 | ssrdv 3827 |
. 2
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓)) → 𝐽 ⊆ 𝐾) |
95 | 13, 94 | impbida 791 |
1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fClus 𝑓) ⊆ (𝐽 fClus 𝑓))) |