Step | Hyp | Ref
| Expression |
1 | | haustop 20945 |
. . 3
⊢ (𝐽 ∈ Haus → 𝐽 ∈ Top) |
2 | | hausflimi 21594 |
. . . 4
⊢ (𝐽 ∈ Haus →
∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) |
3 | 2 | ralrimivw 2950 |
. . 3
⊢ (𝐽 ∈ Haus →
∀𝑓 ∈
(Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) |
4 | 1, 3 | jca 553 |
. 2
⊢ (𝐽 ∈ Haus → (𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))) |
5 | | simpl 472 |
. . . . . . . . . . . . . 14
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝐽 ∈ Top) |
6 | | flimcf.1 |
. . . . . . . . . . . . . . 15
⊢ 𝑋 = ∪
𝐽 |
7 | 6 | toptopon 20548 |
. . . . . . . . . . . . . 14
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋)) |
8 | 5, 7 | sylib 207 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝐽 ∈ (TopOn‘𝑋)) |
9 | | simprll 798 |
. . . . . . . . . . . . . 14
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝑧 ∈ 𝑋) |
10 | 9 | snssd 4281 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑧} ⊆ 𝑋) |
11 | | snnzg 4251 |
. . . . . . . . . . . . . 14
⊢ (𝑧 ∈ 𝑋 → {𝑧} ≠ ∅) |
12 | 9, 11 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑧} ≠ ∅) |
13 | | neifil 21494 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑧} ⊆ 𝑋 ∧ {𝑧} ≠ ∅) → ((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋)) |
14 | 8, 10, 12, 13 | syl3anc 1318 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋)) |
15 | | filfbas 21462 |
. . . . . . . . . . . 12
⊢
(((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑧}) ∈ (fBas‘𝑋)) |
16 | 14, 15 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑧}) ∈ (fBas‘𝑋)) |
17 | | simprlr 799 |
. . . . . . . . . . . . . 14
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝑤 ∈ 𝑋) |
18 | 17 | snssd 4281 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑤} ⊆ 𝑋) |
19 | | snnzg 4251 |
. . . . . . . . . . . . . 14
⊢ (𝑤 ∈ 𝑋 → {𝑤} ≠ ∅) |
20 | 17, 19 | syl 17 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → {𝑤} ≠ ∅) |
21 | | neifil 21494 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ {𝑤} ⊆ 𝑋 ∧ {𝑤} ≠ ∅) → ((nei‘𝐽)‘{𝑤}) ∈ (Fil‘𝑋)) |
22 | 8, 18, 20, 21 | syl3anc 1318 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑤}) ∈ (Fil‘𝑋)) |
23 | | filfbas 21462 |
. . . . . . . . . . . 12
⊢
(((nei‘𝐽)‘{𝑤}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑤}) ∈ (fBas‘𝑋)) |
24 | 22, 23 | syl 17 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑤}) ∈ (fBas‘𝑋)) |
25 | | fbunfip 21483 |
. . . . . . . . . . 11
⊢
((((nei‘𝐽)‘{𝑧}) ∈ (fBas‘𝑋) ∧ ((nei‘𝐽)‘{𝑤}) ∈ (fBas‘𝑋)) → (¬ ∅ ∈
(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ↔ ∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅)) |
26 | 16, 24, 25 | syl2anc 691 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈
(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ↔ ∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅)) |
27 | 6 | neisspw 20721 |
. . . . . . . . . . . . . . 15
⊢ (𝐽 ∈ Top →
((nei‘𝐽)‘{𝑧}) ⊆ 𝒫 𝑋) |
28 | 6 | neisspw 20721 |
. . . . . . . . . . . . . . 15
⊢ (𝐽 ∈ Top →
((nei‘𝐽)‘{𝑤}) ⊆ 𝒫 𝑋) |
29 | 27, 28 | unssd 3751 |
. . . . . . . . . . . . . 14
⊢ (𝐽 ∈ Top →
(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋) |
30 | 29 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋) |
31 | 30 | a1d 25 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈
(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋)) |
32 | | ssun1 3738 |
. . . . . . . . . . . . . 14
⊢
((nei‘𝐽)‘{𝑧}) ⊆ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) |
33 | | filn0 21476 |
. . . . . . . . . . . . . . 15
⊢
(((nei‘𝐽)‘{𝑧}) ∈ (Fil‘𝑋) → ((nei‘𝐽)‘{𝑧}) ≠ ∅) |
34 | 14, 33 | syl 17 |
. . . . . . . . . . . . . 14
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((nei‘𝐽)‘{𝑧}) ≠ ∅) |
35 | | ssn0 3928 |
. . . . . . . . . . . . . 14
⊢
((((nei‘𝐽)‘{𝑧}) ⊆ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ∧ ((nei‘𝐽)‘{𝑧}) ≠ ∅) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅) |
36 | 32, 34, 35 | sylancr 694 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅) |
37 | 36 | a1d 25 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈
(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅)) |
38 | | idd 24 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈
(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → ¬ ∅ ∈
(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) |
39 | 31, 37, 38 | 3jcad 1236 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈
(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → ((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅ ∧ ¬ ∅ ∈
(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) |
40 | 6 | topopn 20536 |
. . . . . . . . . . . . . 14
⊢ (𝐽 ∈ Top → 𝑋 ∈ 𝐽) |
41 | 40 | adantr 480 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → 𝑋 ∈ 𝐽) |
42 | | fsubbas 21481 |
. . . . . . . . . . . . 13
⊢ (𝑋 ∈ 𝐽 → ((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅ ∧ ¬ ∅ ∈
(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) |
43 | 41, 42 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) ↔ ((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅ ∧ ¬ ∅ ∈
(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) |
44 | | fgcl 21492 |
. . . . . . . . . . . . . . 15
⊢
((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋)) |
45 | 44 | adantl 481 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋)) |
46 | | simplrr 797 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑧 ≠ 𝑤) |
47 | 9 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑧 ∈ 𝑋) |
48 | 17 | adantr 480 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑤 ∈ 𝑋) |
49 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((nei‘𝐽)‘{𝑧}) ∈ V |
50 | | fvex 6113 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((nei‘𝐽)‘{𝑤}) ∈ V |
51 | 49, 50 | unex 6854 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ∈ V |
52 | | ssfii 8208 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ∈ V → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) |
53 | 51, 52 | ax-mp 5 |
. . . . . . . . . . . . . . . . . . . 20
⊢
(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) |
54 | | ssfg 21486 |
. . . . . . . . . . . . . . . . . . . . 21
⊢
((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) → (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) |
55 | 54 | adantl 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) |
56 | 53, 55 | syl5ss 3579 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) |
57 | 32, 56 | syl5ss 3579 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → ((nei‘𝐽)‘{𝑧}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) |
58 | 8 | adantr 480 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝐽 ∈ (TopOn‘𝑋)) |
59 | | elflim 21585 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋)) → (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ (𝑧 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑧}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) |
60 | 58, 45, 59 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ (𝑧 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑧}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) |
61 | 47, 57, 60 | mpbir2and 959 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) |
62 | 56 | unssbd 3753 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → ((nei‘𝐽)‘{𝑤}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) |
63 | | elflim 21585 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋)) → (𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ (𝑤 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑤}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) |
64 | 58, 45, 63 | syl2anc 691 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ (𝑤 ∈ 𝑋 ∧ ((nei‘𝐽)‘{𝑤}) ⊆ (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) |
65 | 48, 62, 64 | mpbir2and 959 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) |
66 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 𝑧 → (𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ 𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) |
67 | | eleq1 2676 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 𝑤 → (𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ↔ 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) |
68 | 66, 67 | moi 3356 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ∧ (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ∧ 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) → 𝑧 = 𝑤) |
69 | 68 | 3com23 1263 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ∧ 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) ∧ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) → 𝑧 = 𝑤) |
70 | 69 | 3expia 1259 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ (𝑧 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) ∧ 𝑤 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) → (∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) → 𝑧 = 𝑤)) |
71 | 47, 48, 61, 65, 70 | syl22anc 1319 |
. . . . . . . . . . . . . . . 16
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))) → 𝑧 = 𝑤)) |
72 | 71 | necon3ad 2795 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → (𝑧 ≠ 𝑤 → ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) |
73 | 46, 72 | mpd 15 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) |
74 | | oveq2 6557 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑓 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → (𝐽 fLim 𝑓) = (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) |
75 | 74 | eleq2d 2673 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑓 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → (𝑥 ∈ (𝐽 fLim 𝑓) ↔ 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) |
76 | 75 | mobidv 2479 |
. . . . . . . . . . . . . . . 16
⊢ (𝑓 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → (∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓) ↔ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) |
77 | 76 | notbid 307 |
. . . . . . . . . . . . . . 15
⊢ (𝑓 = (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → (¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓) ↔ ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))))))) |
78 | 77 | rspcev 3282 |
. . . . . . . . . . . . . 14
⊢ (((𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) ∈ (Fil‘𝑋) ∧ ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim (𝑋filGen(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))))) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) |
79 | 45, 73, 78 | syl2anc 691 |
. . . . . . . . . . . . 13
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ (fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋)) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) |
80 | 79 | ex 449 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ((fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) ∈ (fBas‘𝑋) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))) |
81 | 43, 80 | sylbird 249 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (((((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ⊆ 𝒫 𝑋 ∧ (((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})) ≠ ∅ ∧ ¬ ∅ ∈
(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤})))) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))) |
82 | 39, 81 | syld 46 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∅ ∈
(fi‘(((nei‘𝐽)‘{𝑧}) ∪ ((nei‘𝐽)‘{𝑤}))) → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))) |
83 | 26, 82 | sylbird 249 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (∀𝑢 ∈ ((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ → ∃𝑓 ∈ (Fil‘𝑋) ¬ ∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))) |
84 | | df-ne 2782 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∩ 𝑣) ≠ ∅ ↔ ¬ (𝑢 ∩ 𝑣) = ∅) |
85 | 84 | ralbii 2963 |
. . . . . . . . . . . 12
⊢
(∀𝑣 ∈
((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ∀𝑣 ∈ ((nei‘𝐽)‘{𝑤}) ¬ (𝑢 ∩ 𝑣) = ∅) |
86 | | ralnex 2975 |
. . . . . . . . . . . 12
⊢
(∀𝑣 ∈
((nei‘𝐽)‘{𝑤}) ¬ (𝑢 ∩ 𝑣) = ∅ ↔ ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅) |
87 | 85, 86 | bitri 263 |
. . . . . . . . . . 11
⊢
(∀𝑣 ∈
((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅) |
88 | 87 | ralbii 2963 |
. . . . . . . . . 10
⊢
(∀𝑢 ∈
((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ∀𝑢 ∈ ((nei‘𝐽)‘{𝑧}) ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅) |
89 | | ralnex 2975 |
. . . . . . . . . 10
⊢
(∀𝑢 ∈
((nei‘𝐽)‘{𝑧}) ¬ ∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅ ↔ ¬ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅) |
90 | 88, 89 | bitri 263 |
. . . . . . . . 9
⊢
(∀𝑢 ∈
((nei‘𝐽)‘{𝑧})∀𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) ≠ ∅ ↔ ¬ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅) |
91 | | rexnal 2978 |
. . . . . . . . 9
⊢
(∃𝑓 ∈
(Fil‘𝑋) ¬
∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓) ↔ ¬ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) |
92 | 83, 90, 91 | 3imtr3g 283 |
. . . . . . . 8
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (¬ ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅ → ¬ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))) |
93 | 92 | con4d 113 |
. . . . . . 7
⊢ ((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → (∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)) |
94 | 93 | imp 444 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅) |
95 | 94 | an32s 842 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) ∧ ((𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋) ∧ 𝑧 ≠ 𝑤)) → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅) |
96 | 95 | expr 641 |
. . . 4
⊢ (((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) ∧ (𝑧 ∈ 𝑋 ∧ 𝑤 ∈ 𝑋)) → (𝑧 ≠ 𝑤 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)) |
97 | 96 | ralrimivva 2954 |
. . 3
⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (𝑧 ≠ 𝑤 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅)) |
98 | | simpl 472 |
. . . . 5
⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → 𝐽 ∈ Top) |
99 | 98, 7 | sylib 207 |
. . . 4
⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → 𝐽 ∈ (TopOn‘𝑋)) |
100 | | hausnei2 20967 |
. . . 4
⊢ (𝐽 ∈ (TopOn‘𝑋) → (𝐽 ∈ Haus ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (𝑧 ≠ 𝑤 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅))) |
101 | 99, 100 | syl 17 |
. . 3
⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → (𝐽 ∈ Haus ↔ ∀𝑧 ∈ 𝑋 ∀𝑤 ∈ 𝑋 (𝑧 ≠ 𝑤 → ∃𝑢 ∈ ((nei‘𝐽)‘{𝑧})∃𝑣 ∈ ((nei‘𝐽)‘{𝑤})(𝑢 ∩ 𝑣) = ∅))) |
102 | 97, 101 | mpbird 246 |
. 2
⊢ ((𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓)) → 𝐽 ∈ Haus) |
103 | 4, 102 | impbii 198 |
1
⊢ (𝐽 ∈ Haus ↔ (𝐽 ∈ Top ∧ ∀𝑓 ∈ (Fil‘𝑋)∃*𝑥 𝑥 ∈ (𝐽 fLim 𝑓))) |