Step | Hyp | Ref
| Expression |
1 | | cmetmet 23084 |
. . . . . . . . 9
⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (Met‘𝑋)) |
2 | | metxmet 22139 |
. . . . . . . . 9
⊢ (𝐷 ∈ (Met‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
3 | 1, 2 | syl 17 |
. . . . . . . 8
⊢ (𝐷 ∈ (CMet‘𝑋) → 𝐷 ∈ (∞Met‘𝑋)) |
4 | 3 | adantr 481 |
. . . . . . 7
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝐷 ∈ (∞Met‘𝑋)) |
5 | | cmetss.2 |
. . . . . . . 8
⊢ 𝐽 = (MetOpen‘𝐷) |
6 | 5 | mopntopon 22244 |
. . . . . . 7
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ (TopOn‘𝑋)) |
7 | 4, 6 | syl 17 |
. . . . . 6
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝐽 ∈ (TopOn‘𝑋)) |
8 | | resss 5422 |
. . . . . . . 8
⊢ (𝐷 ↾ (𝑌 × 𝑌)) ⊆ 𝐷 |
9 | | dmss 5323 |
. . . . . . . 8
⊢ ((𝐷 ↾ (𝑌 × 𝑌)) ⊆ 𝐷 → dom (𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom 𝐷) |
10 | | dmss 5323 |
. . . . . . . 8
⊢ (dom
(𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom 𝐷 → dom dom (𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom dom 𝐷) |
11 | 8, 9, 10 | mp2b 10 |
. . . . . . 7
⊢ dom dom
(𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom dom 𝐷 |
12 | | cmetmet 23084 |
. . . . . . . . 9
⊢ ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌)) |
13 | | metdmdm 22141 |
. . . . . . . . 9
⊢ ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌) → 𝑌 = dom dom (𝐷 ↾ (𝑌 × 𝑌))) |
14 | 12, 13 | syl 17 |
. . . . . . . 8
⊢ ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) → 𝑌 = dom dom (𝐷 ↾ (𝑌 × 𝑌))) |
15 | | metdmdm 22141 |
. . . . . . . . 9
⊢ (𝐷 ∈ (Met‘𝑋) → 𝑋 = dom dom 𝐷) |
16 | 1, 15 | syl 17 |
. . . . . . . 8
⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 = dom dom 𝐷) |
17 | | sseq12 3628 |
. . . . . . . 8
⊢ ((𝑌 = dom dom (𝐷 ↾ (𝑌 × 𝑌)) ∧ 𝑋 = dom dom 𝐷) → (𝑌 ⊆ 𝑋 ↔ dom dom (𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom dom 𝐷)) |
18 | 14, 16, 17 | syl2anr 495 |
. . . . . . 7
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → (𝑌 ⊆ 𝑋 ↔ dom dom (𝐷 ↾ (𝑌 × 𝑌)) ⊆ dom dom 𝐷)) |
19 | 11, 18 | mpbiri 248 |
. . . . . 6
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑌 ⊆ 𝑋) |
20 | | flimcls 21789 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝑥 ∈ ((cls‘𝐽)‘𝑌) ↔ ∃𝑓 ∈ (Fil‘𝑋)(𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) |
21 | 7, 19, 20 | syl2anc 693 |
. . . . 5
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → (𝑥 ∈ ((cls‘𝐽)‘𝑌) ↔ ∃𝑓 ∈ (Fil‘𝑋)(𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) |
22 | | simprrr 805 |
. . . . . . 7
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑥 ∈ (𝐽 fLim 𝑓)) |
23 | 4 | adantr 481 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐷 ∈ (∞Met‘𝑋)) |
24 | 5 | methaus 22325 |
. . . . . . . . 9
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Haus) |
25 | | hausflimi 21784 |
. . . . . . . . 9
⊢ (𝐽 ∈ Haus →
∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) |
26 | 23, 24, 25 | 3syl 18 |
. . . . . . . 8
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) |
27 | 23, 6 | syl 17 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝐽 ∈ (TopOn‘𝑋)) |
28 | | simprl 794 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (Fil‘𝑋)) |
29 | | simprrl 804 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑌 ∈ 𝑓) |
30 | | flimrest 21787 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝑓) → ((𝐽 ↾t 𝑌) fLim (𝑓 ↾t 𝑌)) = ((𝐽 fLim 𝑓) ∩ 𝑌)) |
31 | 27, 28, 29, 30 | syl3anc 1326 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → ((𝐽 ↾t 𝑌) fLim (𝑓 ↾t 𝑌)) = ((𝐽 fLim 𝑓) ∩ 𝑌)) |
32 | 19 | adantr 481 |
. . . . . . . . . . . . 13
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑌 ⊆ 𝑋) |
33 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢ (𝐷 ↾ (𝑌 × 𝑌)) = (𝐷 ↾ (𝑌 × 𝑌)) |
34 | | eqid 2622 |
. . . . . . . . . . . . . 14
⊢
(MetOpen‘(𝐷
↾ (𝑌 × 𝑌))) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) |
35 | 33, 5, 34 | metrest 22329 |
. . . . . . . . . . . . 13
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐽 ↾t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) |
36 | 23, 32, 35 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝐽 ↾t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) |
37 | 36 | oveq1d 6665 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → ((𝐽 ↾t 𝑌) fLim (𝑓 ↾t 𝑌)) = ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑓 ↾t 𝑌))) |
38 | 31, 37 | eqtr3d 2658 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → ((𝐽 fLim 𝑓) ∩ 𝑌) = ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑓 ↾t 𝑌))) |
39 | | simplr 792 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) |
40 | 5 | flimcfil 23112 |
. . . . . . . . . . . . 13
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑥 ∈ (𝐽 fLim 𝑓)) → 𝑓 ∈ (CauFil‘𝐷)) |
41 | 23, 22, 40 | syl2anc 693 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑓 ∈ (CauFil‘𝐷)) |
42 | | cfilres 23094 |
. . . . . . . . . . . . 13
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (Fil‘𝑋) ∧ 𝑌 ∈ 𝑓) → (𝑓 ∈ (CauFil‘𝐷) ↔ (𝑓 ↾t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))) |
43 | 23, 28, 29, 42 | syl3anc 1326 |
. . . . . . . . . . . 12
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝑓 ∈ (CauFil‘𝐷) ↔ (𝑓 ↾t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌))))) |
44 | 41, 43 | mpbid 222 |
. . . . . . . . . . 11
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝑓 ↾t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) |
45 | 34 | cmetcvg 23083 |
. . . . . . . . . . 11
⊢ (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ∧ (𝑓 ↾t 𝑌) ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑓 ↾t 𝑌)) ≠ ∅) |
46 | 39, 44, 45 | syl2anc 693 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim (𝑓 ↾t 𝑌)) ≠ ∅) |
47 | 38, 46 | eqnetrd 2861 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → ((𝐽 fLim 𝑓) ∩ 𝑌) ≠ ∅) |
48 | | n0 3931 |
. . . . . . . . . 10
⊢ (((𝐽 fLim 𝑓) ∩ 𝑌) ≠ ∅ ↔ ∃𝑥 𝑥 ∈ ((𝐽 fLim 𝑓) ∩ 𝑌)) |
49 | | elin 3796 |
. . . . . . . . . . 11
⊢ (𝑥 ∈ ((𝐽 fLim 𝑓) ∩ 𝑌) ↔ (𝑥 ∈ (𝐽 fLim 𝑓) ∧ 𝑥 ∈ 𝑌)) |
50 | 49 | exbii 1774 |
. . . . . . . . . 10
⊢
(∃𝑥 𝑥 ∈ ((𝐽 fLim 𝑓) ∩ 𝑌) ↔ ∃𝑥(𝑥 ∈ (𝐽 fLim 𝑓) ∧ 𝑥 ∈ 𝑌)) |
51 | 48, 50 | bitri 264 |
. . . . . . . . 9
⊢ (((𝐽 fLim 𝑓) ∩ 𝑌) ≠ ∅ ↔ ∃𝑥(𝑥 ∈ (𝐽 fLim 𝑓) ∧ 𝑥 ∈ 𝑌)) |
52 | 47, 51 | sylib 208 |
. . . . . . . 8
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → ∃𝑥(𝑥 ∈ (𝐽 fLim 𝑓) ∧ 𝑥 ∈ 𝑌)) |
53 | | mopick 2535 |
. . . . . . . 8
⊢
((∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓) ∧ ∃𝑥(𝑥 ∈ (𝐽 fLim 𝑓) ∧ 𝑥 ∈ 𝑌)) → (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥 ∈ 𝑌)) |
54 | 26, 52, 53 | syl2anc 693 |
. . . . . . 7
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → (𝑥 ∈ (𝐽 fLim 𝑓) → 𝑥 ∈ 𝑌)) |
55 | 22, 54 | mpd 15 |
. . . . . 6
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) ∧ (𝑓 ∈ (Fil‘𝑋) ∧ (𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)))) → 𝑥 ∈ 𝑌) |
56 | 55 | rexlimdvaa 3032 |
. . . . 5
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → (∃𝑓 ∈ (Fil‘𝑋)(𝑌 ∈ 𝑓 ∧ 𝑥 ∈ (𝐽 fLim 𝑓)) → 𝑥 ∈ 𝑌)) |
57 | 21, 56 | sylbid 230 |
. . . 4
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → (𝑥 ∈ ((cls‘𝐽)‘𝑌) → 𝑥 ∈ 𝑌)) |
58 | 57 | ssrdv 3609 |
. . 3
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → ((cls‘𝐽)‘𝑌) ⊆ 𝑌) |
59 | 5 | mopntop 22245 |
. . . . 5
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝐽 ∈ Top) |
60 | 4, 59 | syl 17 |
. . . 4
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝐽 ∈ Top) |
61 | 5 | mopnuni 22246 |
. . . . . 6
⊢ (𝐷 ∈ (∞Met‘𝑋) → 𝑋 = ∪ 𝐽) |
62 | 4, 61 | syl 17 |
. . . . 5
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑋 = ∪ 𝐽) |
63 | 19, 62 | sseqtrd 3641 |
. . . 4
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑌 ⊆ ∪ 𝐽) |
64 | | eqid 2622 |
. . . . 5
⊢ ∪ 𝐽 =
∪ 𝐽 |
65 | 64 | iscld4 20869 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ 𝑌 ⊆ ∪ 𝐽)
→ (𝑌 ∈
(Clsd‘𝐽) ↔
((cls‘𝐽)‘𝑌) ⊆ 𝑌)) |
66 | 60, 63, 65 | syl2anc 693 |
. . 3
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → (𝑌 ∈ (Clsd‘𝐽) ↔ ((cls‘𝐽)‘𝑌) ⊆ 𝑌)) |
67 | 58, 66 | mpbird 247 |
. 2
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) → 𝑌 ∈ (Clsd‘𝐽)) |
68 | 1 | adantr 481 |
. . . 4
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐷 ∈ (Met‘𝑋)) |
69 | 64 | cldss 20833 |
. . . . . 6
⊢ (𝑌 ∈ (Clsd‘𝐽) → 𝑌 ⊆ ∪ 𝐽) |
70 | 69 | adantl 482 |
. . . . 5
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ⊆ ∪ 𝐽) |
71 | 68, 2, 61 | 3syl 18 |
. . . . 5
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑋 = ∪ 𝐽) |
72 | 70, 71 | sseqtr4d 3642 |
. . . 4
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝑌 ⊆ 𝑋) |
73 | | metres2 22168 |
. . . 4
⊢ ((𝐷 ∈ (Met‘𝑋) ∧ 𝑌 ⊆ 𝑋) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌)) |
74 | 68, 72, 73 | syl2anc 693 |
. . 3
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌)) |
75 | 3 | ad2antrr 762 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐷 ∈ (∞Met‘𝑋)) |
76 | 72 | adantr 481 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑌 ⊆ 𝑋) |
77 | 75, 76, 35 | syl2anc 693 |
. . . . . . . 8
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽 ↾t 𝑌) = (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌)))) |
78 | 77 | eqcomd 2628 |
. . . . . . 7
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) = (𝐽 ↾t 𝑌)) |
79 | | metxmet 22139 |
. . . . . . . . . . 11
⊢ ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌)) |
80 | 74, 79 | syl 17 |
. . . . . . . . . 10
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌)) |
81 | | cfilfil 23065 |
. . . . . . . . . 10
⊢ (((𝐷 ↾ (𝑌 × 𝑌)) ∈ (∞Met‘𝑌) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (Fil‘𝑌)) |
82 | 80, 81 | sylan 488 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (Fil‘𝑌)) |
83 | | elfvdm 6220 |
. . . . . . . . . 10
⊢ (𝐷 ∈ (CMet‘𝑋) → 𝑋 ∈ dom CMet) |
84 | 83 | ad2antrr 762 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑋 ∈ dom CMet) |
85 | | trfg 21695 |
. . . . . . . . 9
⊢ ((𝑓 ∈ (Fil‘𝑌) ∧ 𝑌 ⊆ 𝑋 ∧ 𝑋 ∈ dom CMet) → ((𝑋filGen𝑓) ↾t 𝑌) = 𝑓) |
86 | 82, 76, 84, 85 | syl3anc 1326 |
. . . . . . . 8
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((𝑋filGen𝑓) ↾t 𝑌) = 𝑓) |
87 | 86 | eqcomd 2628 |
. . . . . . 7
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 = ((𝑋filGen𝑓) ↾t 𝑌)) |
88 | 78, 87 | oveq12d 6668 |
. . . . . 6
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) = ((𝐽 ↾t 𝑌) fLim ((𝑋filGen𝑓) ↾t 𝑌))) |
89 | 75, 6 | syl 17 |
. . . . . . 7
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐽 ∈ (TopOn‘𝑋)) |
90 | | filfbas 21652 |
. . . . . . . . . 10
⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ∈ (fBas‘𝑌)) |
91 | 82, 90 | syl 17 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (fBas‘𝑌)) |
92 | | filsspw 21655 |
. . . . . . . . . . 11
⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑓 ⊆ 𝒫 𝑌) |
93 | 82, 92 | syl 17 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ⊆ 𝒫 𝑌) |
94 | | sspwb 4917 |
. . . . . . . . . . 11
⊢ (𝑌 ⊆ 𝑋 ↔ 𝒫 𝑌 ⊆ 𝒫 𝑋) |
95 | 76, 94 | sylib 208 |
. . . . . . . . . 10
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝒫 𝑌 ⊆ 𝒫 𝑋) |
96 | 93, 95 | sstrd 3613 |
. . . . . . . . 9
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ⊆ 𝒫 𝑋) |
97 | | fbasweak 21669 |
. . . . . . . . 9
⊢ ((𝑓 ∈ (fBas‘𝑌) ∧ 𝑓 ⊆ 𝒫 𝑋 ∧ 𝑋 ∈ dom CMet) → 𝑓 ∈ (fBas‘𝑋)) |
98 | 91, 96, 84, 97 | syl3anc 1326 |
. . . . . . . 8
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ∈ (fBas‘𝑋)) |
99 | | fgcl 21682 |
. . . . . . . 8
⊢ (𝑓 ∈ (fBas‘𝑋) → (𝑋filGen𝑓) ∈ (Fil‘𝑋)) |
100 | 98, 99 | syl 17 |
. . . . . . 7
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝑓) ∈ (Fil‘𝑋)) |
101 | | ssfg 21676 |
. . . . . . . . 9
⊢ (𝑓 ∈ (fBas‘𝑋) → 𝑓 ⊆ (𝑋filGen𝑓)) |
102 | 98, 101 | syl 17 |
. . . . . . . 8
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑓 ⊆ (𝑋filGen𝑓)) |
103 | | filtop 21659 |
. . . . . . . . 9
⊢ (𝑓 ∈ (Fil‘𝑌) → 𝑌 ∈ 𝑓) |
104 | 82, 103 | syl 17 |
. . . . . . . 8
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑌 ∈ 𝑓) |
105 | 102, 104 | sseldd 3604 |
. . . . . . 7
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝑌 ∈ (𝑋filGen𝑓)) |
106 | | flimrest 21787 |
. . . . . . 7
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋filGen𝑓) ∈ (Fil‘𝑋) ∧ 𝑌 ∈ (𝑋filGen𝑓)) → ((𝐽 ↾t 𝑌) fLim ((𝑋filGen𝑓) ↾t 𝑌)) = ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌)) |
107 | 89, 100, 105, 106 | syl3anc 1326 |
. . . . . 6
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((𝐽 ↾t 𝑌) fLim ((𝑋filGen𝑓) ↾t 𝑌)) = ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌)) |
108 | | flimclsi 21782 |
. . . . . . . . 9
⊢ (𝑌 ∈ (𝑋filGen𝑓) → (𝐽 fLim (𝑋filGen𝑓)) ⊆ ((cls‘𝐽)‘𝑌)) |
109 | 105, 108 | syl 17 |
. . . . . . . 8
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽 fLim (𝑋filGen𝑓)) ⊆ ((cls‘𝐽)‘𝑌)) |
110 | | cldcls 20846 |
. . . . . . . . 9
⊢ (𝑌 ∈ (Clsd‘𝐽) → ((cls‘𝐽)‘𝑌) = 𝑌) |
111 | 110 | ad2antlr 763 |
. . . . . . . 8
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((cls‘𝐽)‘𝑌) = 𝑌) |
112 | 109, 111 | sseqtrd 3641 |
. . . . . . 7
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽 fLim (𝑋filGen𝑓)) ⊆ 𝑌) |
113 | | df-ss 3588 |
. . . . . . 7
⊢ ((𝐽 fLim (𝑋filGen𝑓)) ⊆ 𝑌 ↔ ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌) = (𝐽 fLim (𝑋filGen𝑓))) |
114 | 112, 113 | sylib 208 |
. . . . . 6
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((𝐽 fLim (𝑋filGen𝑓)) ∩ 𝑌) = (𝐽 fLim (𝑋filGen𝑓))) |
115 | 88, 107, 114 | 3eqtrd 2660 |
. . . . 5
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) = (𝐽 fLim (𝑋filGen𝑓))) |
116 | | simpll 790 |
. . . . . 6
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → 𝐷 ∈ (CMet‘𝑋)) |
117 | 68, 2 | syl 17 |
. . . . . . 7
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → 𝐷 ∈ (∞Met‘𝑋)) |
118 | | cfilresi 23093 |
. . . . . . 7
⊢ ((𝐷 ∈ (∞Met‘𝑋) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝑓) ∈ (CauFil‘𝐷)) |
119 | 117, 118 | sylan 488 |
. . . . . 6
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝑋filGen𝑓) ∈ (CauFil‘𝐷)) |
120 | 5 | cmetcvg 23083 |
. . . . . 6
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ (𝑋filGen𝑓) ∈ (CauFil‘𝐷)) → (𝐽 fLim (𝑋filGen𝑓)) ≠ ∅) |
121 | 116, 119,
120 | syl2anc 693 |
. . . . 5
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → (𝐽 fLim (𝑋filGen𝑓)) ≠ ∅) |
122 | 115, 121 | eqnetrd 2861 |
. . . 4
⊢ (((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) ∧ 𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))) → ((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) ≠ ∅) |
123 | 122 | ralrimiva 2966 |
. . 3
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → ∀𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) ≠ ∅) |
124 | 34 | iscmet 23082 |
. . 3
⊢ ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ↔ ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (Met‘𝑌) ∧ ∀𝑓 ∈ (CauFil‘(𝐷 ↾ (𝑌 × 𝑌)))((MetOpen‘(𝐷 ↾ (𝑌 × 𝑌))) fLim 𝑓) ≠ ∅)) |
125 | 74, 123, 124 | sylanbrc 698 |
. 2
⊢ ((𝐷 ∈ (CMet‘𝑋) ∧ 𝑌 ∈ (Clsd‘𝐽)) → (𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌)) |
126 | 67, 125 | impbida 877 |
1
⊢ (𝐷 ∈ (CMet‘𝑋) → ((𝐷 ↾ (𝑌 × 𝑌)) ∈ (CMet‘𝑌) ↔ 𝑌 ∈ (Clsd‘𝐽))) |