Step | Hyp | Ref
| Expression |
1 | | simplll 773 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝐽 ∈ (TopOn‘𝑋)) |
2 | | simprl 769 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝑓 ∈ (Fil‘𝑋)) |
3 | | simplr 767 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝐽 ⊆ 𝐾) |
4 | | flimss1 22581 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) |
5 | 1, 2, 3, 4 | syl3anc 1367 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → (𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) |
6 | | simprr 771 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝑥 ∈ (𝐾 fLim 𝑓)) |
7 | 5, 6 | sseldd 3968 |
. . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ 𝑥 ∈ (𝐾 fLim 𝑓))) → 𝑥 ∈ (𝐽 fLim 𝑓)) |
8 | 7 | expr 459 |
. . . 4
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (Fil‘𝑋)) → (𝑥 ∈ (𝐾 fLim 𝑓) → 𝑥 ∈ (𝐽 fLim 𝑓))) |
9 | 8 | ssrdv 3973 |
. . 3
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) ∧ 𝑓 ∈ (Fil‘𝑋)) → (𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) |
10 | 9 | ralrimiva 3182 |
. 2
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ 𝐽 ⊆ 𝐾) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) |
11 | | oveq2 7164 |
. . . . . . . . . . . 12
⊢ (𝑓 = ((nei‘𝐾)‘{𝑦}) → (𝐾 fLim 𝑓) = (𝐾 fLim ((nei‘𝐾)‘{𝑦}))) |
12 | | oveq2 7164 |
. . . . . . . . . . . 12
⊢ (𝑓 = ((nei‘𝐾)‘{𝑦}) → (𝐽 fLim 𝑓) = (𝐽 fLim ((nei‘𝐾)‘{𝑦}))) |
13 | 11, 12 | sseq12d 4000 |
. . . . . . . . . . 11
⊢ (𝑓 = ((nei‘𝐾)‘{𝑦}) → ((𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓) ↔ (𝐾 fLim ((nei‘𝐾)‘{𝑦})) ⊆ (𝐽 fLim ((nei‘𝐾)‘{𝑦})))) |
14 | | simplr 767 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) |
15 | | simpllr 774 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐾 ∈ (TopOn‘𝑋)) |
16 | | simplll 773 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ (TopOn‘𝑋)) |
17 | | simprl 769 |
. . . . . . . . . . . . . . 15
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ 𝐽) |
18 | | toponss 21535 |
. . . . . . . . . . . . . . 15
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑥 ∈ 𝐽) → 𝑥 ⊆ 𝑋) |
19 | 16, 17, 18 | syl2anc 586 |
. . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ⊆ 𝑋) |
20 | | simprr 771 |
. . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑥) |
21 | 19, 20 | sseldd 3968 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ 𝑋) |
22 | 21 | snssd 4742 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑦} ⊆ 𝑋) |
23 | | snnzg 4710 |
. . . . . . . . . . . . 13
⊢ (𝑦 ∈ 𝑋 → {𝑦} ≠ ∅) |
24 | 21, 23 | syl 17 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → {𝑦} ≠ ∅) |
25 | | neifil 22488 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ {𝑦} ⊆ 𝑋 ∧ {𝑦} ≠ ∅) → ((nei‘𝐾)‘{𝑦}) ∈ (Fil‘𝑋)) |
26 | 15, 22, 24, 25 | syl3anc 1367 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ((nei‘𝐾)‘{𝑦}) ∈ (Fil‘𝑋)) |
27 | 13, 14, 26 | rspcdva 3625 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → (𝐾 fLim ((nei‘𝐾)‘{𝑦})) ⊆ (𝐽 fLim ((nei‘𝐾)‘{𝑦}))) |
28 | | neiflim 22582 |
. . . . . . . . . . 11
⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝑦 ∈ 𝑋) → 𝑦 ∈ (𝐾 fLim ((nei‘𝐾)‘{𝑦}))) |
29 | 15, 21, 28 | syl2anc 586 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ (𝐾 fLim ((nei‘𝐾)‘{𝑦}))) |
30 | 27, 29 | sseldd 3968 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑦 ∈ (𝐽 fLim ((nei‘𝐾)‘{𝑦}))) |
31 | | flimneiss 22574 |
. . . . . . . . 9
⊢ (𝑦 ∈ (𝐽 fLim ((nei‘𝐾)‘{𝑦})) → ((nei‘𝐽)‘{𝑦}) ⊆ ((nei‘𝐾)‘{𝑦})) |
32 | 30, 31 | syl 17 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → ((nei‘𝐽)‘{𝑦}) ⊆ ((nei‘𝐾)‘{𝑦})) |
33 | | topontop 21521 |
. . . . . . . . . 10
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) |
34 | 16, 33 | syl 17 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝐽 ∈ Top) |
35 | | opnneip 21727 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ ((nei‘𝐽)‘{𝑦})) |
36 | 34, 17, 20, 35 | syl3anc 1367 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ ((nei‘𝐽)‘{𝑦})) |
37 | 32, 36 | sseldd 3968 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝑥)) → 𝑥 ∈ ((nei‘𝐾)‘{𝑦})) |
38 | 37 | anassrs 470 |
. . . . . 6
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥 ∈ 𝐽) ∧ 𝑦 ∈ 𝑥) → 𝑥 ∈ ((nei‘𝐾)‘{𝑦})) |
39 | 38 | ralrimiva 3182 |
. . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥 ∈ 𝐽) → ∀𝑦 ∈ 𝑥 𝑥 ∈ ((nei‘𝐾)‘{𝑦})) |
40 | | simpllr 774 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥 ∈ 𝐽) → 𝐾 ∈ (TopOn‘𝑋)) |
41 | | topontop 21521 |
. . . . . 6
⊢ (𝐾 ∈ (TopOn‘𝑋) → 𝐾 ∈ Top) |
42 | | opnnei 21728 |
. . . . . 6
⊢ (𝐾 ∈ Top → (𝑥 ∈ 𝐾 ↔ ∀𝑦 ∈ 𝑥 𝑥 ∈ ((nei‘𝐾)‘{𝑦}))) |
43 | 40, 41, 42 | 3syl 18 |
. . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥 ∈ 𝐽) → (𝑥 ∈ 𝐾 ↔ ∀𝑦 ∈ 𝑥 𝑥 ∈ ((nei‘𝐾)‘{𝑦}))) |
44 | 39, 43 | mpbird 259 |
. . . 4
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) ∧ 𝑥 ∈ 𝐽) → 𝑥 ∈ 𝐾) |
45 | 44 | ex 415 |
. . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) → (𝑥 ∈ 𝐽 → 𝑥 ∈ 𝐾)) |
46 | 45 | ssrdv 3973 |
. 2
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) ∧ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓)) → 𝐽 ⊆ 𝐾) |
47 | 10, 46 | impbida 799 |
1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑋)) → (𝐽 ⊆ 𝐾 ↔ ∀𝑓 ∈ (Fil‘𝑋)(𝐾 fLim 𝑓) ⊆ (𝐽 fLim 𝑓))) |