Step | Hyp | Ref
| Expression |
1 | | haustop 21554 |
. . 3
⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top) |
2 | | cnextfrel.1 |
. . . 4
⊢ 𝐶 = ∪
𝐽 |
3 | | cnextfrel.2 |
. . . 4
⊢ 𝐵 = ∪
𝐾 |
4 | 2, 3 | cnextrel 22286 |
. . 3
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Rel ((𝐽CnExt𝐾)‘𝐹)) |
5 | 1, 4 | sylanl2 671 |
. 2
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Rel ((𝐽CnExt𝐾)‘𝐹)) |
6 | | simpllr 766 |
. . . . . . 7
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝐾 ∈ Haus) |
7 | 2 | toptopon 21140 |
. . . . . . . . . 10
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶)) |
8 | 7 | biimpi 208 |
. . . . . . . . 9
⊢ (𝐽 ∈ Top → 𝐽 ∈ (TopOn‘𝐶)) |
9 | 8 | ad3antrrr 720 |
. . . . . . . 8
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝐽 ∈ (TopOn‘𝐶)) |
10 | | simplrr 768 |
. . . . . . . 8
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝐴 ⊆ 𝐶) |
11 | 9, 7 | sylibr 226 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝐽 ∈ Top) |
12 | 2 | clsss3 21282 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝐴 ⊆ 𝐶) → ((cls‘𝐽)‘𝐴) ⊆ 𝐶) |
13 | 11, 10, 12 | syl2anc 579 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ((cls‘𝐽)‘𝐴) ⊆ 𝐶) |
14 | | simpr 479 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝑥 ∈ ((cls‘𝐽)‘𝐴)) |
15 | 13, 14 | sseldd 3822 |
. . . . . . . 8
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝑥 ∈ 𝐶) |
16 | | trnei 22115 |
. . . . . . . . 9
⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))) |
17 | 16 | biimpa 470 |
. . . . . . . 8
⊢ (((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐶) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)) |
18 | 9, 10, 15, 14, 17 | syl31anc 1441 |
. . . . . . 7
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)) |
19 | | simplrl 767 |
. . . . . . 7
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → 𝐹:𝐴⟶𝐵) |
20 | 3 | hausflf 22220 |
. . . . . . 7
⊢ ((𝐾 ∈ Haus ∧
(((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝐵) → ∃*𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) |
21 | 6, 18, 19, 20 | syl3anc 1439 |
. . . . . 6
⊢ ((((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) ∧ 𝑥 ∈ ((cls‘𝐽)‘𝐴)) → ∃*𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) |
22 | 21 | ex 403 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) → ∃*𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
23 | 22 | alrimiv 1970 |
. . . 4
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → ∀𝑥(𝑥 ∈ ((cls‘𝐽)‘𝐴) → ∃*𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
24 | | moanimv 2654 |
. . . . 5
⊢
(∃*𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) → ∃*𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
25 | 24 | albii 1863 |
. . . 4
⊢
(∀𝑥∃*𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ ∀𝑥(𝑥 ∈ ((cls‘𝐽)‘𝐴) → ∃*𝑦 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
26 | 23, 25 | sylibr 226 |
. . 3
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → ∀𝑥∃*𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
27 | | df-br 4889 |
. . . . . . 7
⊢ (𝑥((𝐽CnExt𝐾)‘𝐹)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ((𝐽CnExt𝐾)‘𝐹)) |
28 | 27 | a1i 11 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → (𝑥((𝐽CnExt𝐾)‘𝐹)𝑦 ↔ 〈𝑥, 𝑦〉 ∈ ((𝐽CnExt𝐾)‘𝐹))) |
29 | 2, 3 | cnextfval 22285 |
. . . . . . . 8
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Top) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → ((𝐽CnExt𝐾)‘𝐹) = ∪
𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
30 | 1, 29 | sylanl2 671 |
. . . . . . 7
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → ((𝐽CnExt𝐾)‘𝐹) = ∪
𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
31 | 30 | eleq2d 2845 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → (〈𝑥, 𝑦〉 ∈ ((𝐽CnExt𝐾)‘𝐹) ↔ 〈𝑥, 𝑦〉 ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))) |
32 | | opeliunxp 5418 |
. . . . . . 7
⊢
(〈𝑥, 𝑦〉 ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹))) |
33 | 32 | a1i 11 |
. . . . . 6
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → (〈𝑥, 𝑦〉 ∈ ∪ 𝑥 ∈ ((cls‘𝐽)‘𝐴)({𝑥} × ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))) |
34 | 28, 31, 33 | 3bitrd 297 |
. . . . 5
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → (𝑥((𝐽CnExt𝐾)‘𝐹)𝑦 ↔ (𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))) |
35 | 34 | mobidv 2564 |
. . . 4
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → (∃*𝑦 𝑥((𝐽CnExt𝐾)‘𝐹)𝑦 ↔ ∃*𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))) |
36 | 35 | albidv 1963 |
. . 3
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → (∀𝑥∃*𝑦 𝑥((𝐽CnExt𝐾)‘𝐹)𝑦 ↔ ∀𝑥∃*𝑦(𝑥 ∈ ((cls‘𝐽)‘𝐴) ∧ 𝑦 ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)))) |
37 | 26, 36 | mpbird 249 |
. 2
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → ∀𝑥∃*𝑦 𝑥((𝐽CnExt𝐾)‘𝐹)𝑦) |
38 | | dffun6 6152 |
. 2
⊢ (Fun
((𝐽CnExt𝐾)‘𝐹) ↔ (Rel ((𝐽CnExt𝐾)‘𝐹) ∧ ∀𝑥∃*𝑦 𝑥((𝐽CnExt𝐾)‘𝐹)𝑦)) |
39 | 5, 37, 38 | sylanbrc 578 |
1
⊢ (((𝐽 ∈ Top ∧ 𝐾 ∈ Haus) ∧ (𝐹:𝐴⟶𝐵 ∧ 𝐴 ⊆ 𝐶)) → Fun ((𝐽CnExt𝐾)‘𝐹)) |