Step | Hyp | Ref
| Expression |
1 | | simpll 767 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝜑) |
2 | | simpll 767 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → 𝜑) |
3 | | simpr3 1197 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) |
4 | | cnextf.3 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐽 ∈ Top) |
5 | 4 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → 𝐽 ∈ Top) |
6 | | simpr2 1196 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → 𝑑 ∈ ((nei‘𝐽)‘{𝑥})) |
7 | | neii2 21861 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥})) → ∃𝑣 ∈ 𝐽 ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) |
8 | 5, 6, 7 | syl2anc 587 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → ∃𝑣 ∈ 𝐽 ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) |
9 | | vex 3402 |
. . . . . . . . . . . . . . . . . . . 20
⊢ 𝑥 ∈ V |
10 | 9 | snss 4674 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 ∈ 𝑣 ↔ {𝑥} ⊆ 𝑣) |
11 | 10 | biimpri 231 |
. . . . . . . . . . . . . . . . . 18
⊢ ({𝑥} ⊆ 𝑣 → 𝑥 ∈ 𝑣) |
12 | 11 | anim1i 618 |
. . . . . . . . . . . . . . . . 17
⊢ (({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑) → (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)) |
13 | 12 | anim2i 620 |
. . . . . . . . . . . . . . . 16
⊢ ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))) |
14 | 13 | anim2i 620 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑))) → (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)))) |
15 | 14 | ex 416 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))))) |
16 | | 3anass 1096 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ↔ (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣))) |
17 | 16 | anbi1i 627 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) ↔ ((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)) ∧ 𝑣 ⊆ 𝑑)) |
18 | | anass 472 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)) ∧ 𝑣 ⊆ 𝑑) ↔ (𝜑 ∧ ((𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑))) |
19 | | anass 472 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) ↔ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))) |
20 | 19 | anbi2i 626 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ ((𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑)) ↔ (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)))) |
21 | 17, 18, 20 | 3bitri 300 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) ↔ (𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑)))) |
22 | | opnneip 21872 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) |
23 | 4, 22 | syl3an1 1164 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) |
24 | 23 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) → 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) |
25 | | simpr2 1196 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝑣 ⊆ 𝑑 ∧ (𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣)) → 𝑣 ∈ 𝐽) |
26 | 25 | ex 416 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑣 ⊆ 𝑑 → ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) → 𝑣 ∈ 𝐽)) |
27 | 26 | imdistanri 573 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) → (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑)) |
28 | 24, 27 | jca 515 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑥 ∈ 𝑣) ∧ 𝑣 ⊆ 𝑑) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑))) |
29 | 21, 28 | sylbir 238 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ (𝑣 ∈ 𝐽 ∧ (𝑥 ∈ 𝑣 ∧ 𝑣 ⊆ 𝑑))) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑))) |
30 | 15, 29 | syl6 35 |
. . . . . . . . . . . . 13
⊢ (𝜑 → ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑)))) |
31 | 30 | adantr 484 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑)))) |
32 | | cnextf.4 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → 𝐾 ∈ Haus) |
33 | | haustop 22084 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐾 ∈ Haus → 𝐾 ∈ Top) |
34 | 32, 33 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → 𝐾 ∈ Top) |
35 | | imassrn 5914 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝐹 “ (𝑑 ∩ 𝐴)) ⊆ ran 𝐹 |
36 | | cnextf.5 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
37 | 36 | frnd 6512 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝜑 → ran 𝐹 ⊆ 𝐵) |
38 | 35, 37 | sstrid 3888 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝜑 → (𝐹 “ (𝑑 ∩ 𝐴)) ⊆ 𝐵) |
39 | | ssrin 4124 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑣 ⊆ 𝑑 → (𝑣 ∩ 𝐴) ⊆ (𝑑 ∩ 𝐴)) |
40 | | imass2 5939 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝑣 ∩ 𝐴) ⊆ (𝑑 ∩ 𝐴) → (𝐹 “ (𝑣 ∩ 𝐴)) ⊆ (𝐹 “ (𝑑 ∩ 𝐴))) |
41 | 39, 40 | syl 17 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑣 ⊆ 𝑑 → (𝐹 “ (𝑣 ∩ 𝐴)) ⊆ (𝐹 “ (𝑑 ∩ 𝐴))) |
42 | | cnextf.2 |
. . . . . . . . . . . . . . . . . . 19
⊢ 𝐵 = ∪
𝐾 |
43 | 42 | clsss 21807 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐾 ∈ Top ∧ (𝐹 “ (𝑑 ∩ 𝐴)) ⊆ 𝐵 ∧ (𝐹 “ (𝑣 ∩ 𝐴)) ⊆ (𝐹 “ (𝑑 ∩ 𝐴))) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴)))) |
44 | 34, 38, 41, 43 | syl2an3an 1423 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑣 ⊆ 𝑑) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴)))) |
45 | | sstr 3885 |
. . . . . . . . . . . . . . . . 17
⊢
((((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) |
46 | 44, 45 | sylan 583 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑣 ⊆ 𝑑) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) |
47 | 46 | an32s 652 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑣 ⊆ 𝑑) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) |
48 | 47 | ex 416 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → (𝑣 ⊆ 𝑑 → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)) |
49 | 48 | anim2d 615 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑) → (𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))) |
50 | 49 | anim2d 615 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ 𝑣 ⊆ 𝑑)) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)))) |
51 | 31, 50 | syld 47 |
. . . . . . . . . . 11
⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ((𝑣 ∈ 𝐽 ∧ ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → (𝑣 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)))) |
52 | 51 | reximdv2 3181 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → (∃𝑣 ∈ 𝐽 ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤))) |
53 | 52 | imp 410 |
. . . . . . . . 9
⊢ (((𝜑 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) ∧ ∃𝑣 ∈ 𝐽 ({𝑥} ⊆ 𝑣 ∧ 𝑣 ⊆ 𝑑)) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)) |
54 | 2, 3, 8, 53 | syl21anc 837 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ (𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥}) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)) |
55 | 54 | 3anassrs 1361 |
. . . . . . 7
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑑 ∈ ((nei‘𝐽)‘{𝑥})) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)) |
56 | | simpr 488 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∧ ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) |
57 | | simp-4l 783 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∧ ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) → 𝜑) |
58 | | simplr 769 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∧ ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) → 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) |
59 | | imaeq2 5899 |
. . . . . . . . . . . . . 14
⊢ (𝑢 = (𝑑 ∩ 𝐴) → (𝐹 “ 𝑢) = (𝐹 “ (𝑑 ∩ 𝐴))) |
60 | 59 | fveq2d 6680 |
. . . . . . . . . . . . 13
⊢ (𝑢 = (𝑑 ∩ 𝐴) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) = ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴)))) |
61 | 60 | sseq1d 3908 |
. . . . . . . . . . . 12
⊢ (𝑢 = (𝑑 ∩ 𝐴) → (((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤 ↔ ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) |
62 | 61 | biimpcd 252 |
. . . . . . . . . . 11
⊢
(((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤 → (𝑢 = (𝑑 ∩ 𝐴) → ((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) |
63 | 62 | reximdv 3183 |
. . . . . . . . . 10
⊢
(((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤 → (∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})𝑢 = (𝑑 ∩ 𝐴) → ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤)) |
64 | | fvexd 6691 |
. . . . . . . . . . . 12
⊢ (𝜑 → ((nei‘𝐽)‘{𝑥}) ∈ V) |
65 | | cnextf.1 |
. . . . . . . . . . . . . . . 16
⊢ 𝐶 = ∪
𝐽 |
66 | 65 | toptopon 21670 |
. . . . . . . . . . . . . . 15
⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝐶)) |
67 | 4, 66 | sylib 221 |
. . . . . . . . . . . . . 14
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝐶)) |
68 | 67 | elfvexd 6710 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐶 ∈ V) |
69 | | cnextf.a |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
70 | 68, 69 | ssexd 5192 |
. . . . . . . . . . . 12
⊢ (𝜑 → 𝐴 ∈ V) |
71 | | elrest 16806 |
. . . . . . . . . . . 12
⊢
((((nei‘𝐽)‘{𝑥}) ∈ V ∧ 𝐴 ∈ V) → (𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↔ ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})𝑢 = (𝑑 ∩ 𝐴))) |
72 | 64, 70, 71 | syl2anc 587 |
. . . . . . . . . . 11
⊢ (𝜑 → (𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ↔ ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})𝑢 = (𝑑 ∩ 𝐴))) |
73 | 72 | biimpa 480 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) → ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})𝑢 = (𝑑 ∩ 𝐴)) |
74 | 63, 73 | impel 509 |
. . . . . . . . 9
⊢
((((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤 ∧ (𝜑 ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))) → ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) |
75 | 56, 57, 58, 74 | syl12anc 836 |
. . . . . . . 8
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) ∧ ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) → ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) |
76 | | cnextf.6 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → ((cls‘𝐽)‘𝐴) = 𝐶) |
77 | | eleq1w 2815 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐶 ↔ 𝑦 ∈ 𝐶)) |
78 | 77 | anbi2d 632 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → ((𝜑 ∧ 𝑥 ∈ 𝐶) ↔ (𝜑 ∧ 𝑦 ∈ 𝐶))) |
79 | | sneq 4526 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 𝑦 → {𝑥} = {𝑦}) |
80 | 79 | fveq2d 6680 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑦 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑦})) |
81 | 80 | oveq1d 7187 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑦 → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) = (((nei‘𝐽)‘{𝑦}) ↾t 𝐴)) |
82 | 81 | oveq2d 7188 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 𝑦 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾t 𝐴))) |
83 | 82 | fveq1d 6678 |
. . . . . . . . . . . . . . . . . . 19
⊢ (𝑥 = 𝑦 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾t 𝐴))‘𝐹)) |
84 | 83 | neeq1d 2993 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑥 = 𝑦 → (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅ ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾t 𝐴))‘𝐹) ≠ ∅)) |
85 | 78, 84 | imbi12d 348 |
. . . . . . . . . . . . . . . . 17
⊢ (𝑥 = 𝑦 → (((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅) ↔ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾t 𝐴))‘𝐹) ≠ ∅))) |
86 | | cnextf.7 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅) |
87 | 85, 86 | chvarvv 2010 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑦 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑦}) ↾t 𝐴))‘𝐹) ≠ ∅) |
88 | 65, 42, 4, 32, 36, 69, 76, 87 | cnextfvval 22818 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) |
89 | | fvex 6689 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ∈ V |
90 | 89 | uniex 7487 |
. . . . . . . . . . . . . . . . 17
⊢ ∪ ((𝐾
fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ∈ V |
91 | 90 | snid 4552 |
. . . . . . . . . . . . . . . 16
⊢ ∪ ((𝐾
fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ∈ {∪
((𝐾 fLimf
(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)} |
92 | 32 | adantr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐾 ∈ Haus) |
93 | 76 | eleq2d 2818 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝜑 → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑥 ∈ 𝐶)) |
94 | 93 | biimpar 481 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ ((cls‘𝐽)‘𝐴)) |
95 | 67 | adantr 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐽 ∈ (TopOn‘𝐶)) |
96 | 69 | adantr 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐴 ⊆ 𝐶) |
97 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝑥 ∈ 𝐶) |
98 | | trnei 22645 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))) |
99 | 95, 96, 97, 98 | syl3anc 1372 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (𝑥 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴))) |
100 | 94, 99 | mpbid 235 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴)) |
101 | 36 | adantr 484 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐹:𝐴⟶𝐵) |
102 | 42 | hausflf2 22751 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐾 ∈ Haus ∧
(((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝐵) ∧ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1o) |
103 | 92, 100, 101, 86, 102 | syl31anc 1374 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1o) |
104 | | en1b 8626 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≈ 1o ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)}) |
105 | 103, 104 | sylib 221 |
. . . . . . . . . . . . . . . 16
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)}) |
106 | 91, 105 | eleqtrrid 2840 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∪
((𝐾 fLimf
(((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) |
107 | 88, 106 | eqeltrd 2833 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹)) |
108 | 42 | toptopon 21670 |
. . . . . . . . . . . . . . . . 17
⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝐵)) |
109 | 34, 108 | sylib 221 |
. . . . . . . . . . . . . . . 16
⊢ (𝜑 → 𝐾 ∈ (TopOn‘𝐵)) |
110 | 109 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → 𝐾 ∈ (TopOn‘𝐵)) |
111 | | flfnei 22744 |
. . . . . . . . . . . . . . 15
⊢ ((𝐾 ∈ (TopOn‘𝐵) ∧ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝐵) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ↔ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝐵 ∧ ∀𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏))) |
112 | 110, 100,
101, 111 | syl3anc 1372 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ↔ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝐵 ∧ ∀𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏))) |
113 | 107, 112 | mpbid 235 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝐵 ∧ ∀𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏)) |
114 | 113 | simprd 499 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∀𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏) |
115 | 114 | r19.21bi 3121 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏) |
116 | 115 | ad4ant13 751 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏) |
117 | 34 | ad3antrrr 730 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝐾 ∈ Top) |
118 | | simplr 769 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) |
119 | 42 | neii1 21859 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ Top ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝑏 ⊆ 𝐵) |
120 | 117, 118,
119 | syl2anc 587 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → 𝑏 ⊆ 𝐵) |
121 | | simpr 488 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘𝑏) ⊆ 𝑤) |
122 | 42 | clsss 21807 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ (𝐹 “ 𝑢) ⊆ 𝑏) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ ((cls‘𝐾)‘𝑏)) |
123 | | sstr 3885 |
. . . . . . . . . . . . . . . 16
⊢
((((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ ((cls‘𝐾)‘𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) |
124 | 122, 123 | sylan 583 |
. . . . . . . . . . . . . . 15
⊢ (((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ (𝐹 “ 𝑢) ⊆ 𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) |
125 | 124 | 3an1rs 1360 |
. . . . . . . . . . . . . 14
⊢ (((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) ∧ (𝐹 “ 𝑢) ⊆ 𝑏) → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) |
126 | 125 | ex 416 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ((𝐹 “ 𝑢) ⊆ 𝑏 → ((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)) |
127 | 126 | reximdv 3183 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Top ∧ 𝑏 ⊆ 𝐵 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏 → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)) |
128 | 117, 120,
121, 127 | syl3anc 1372 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏 → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)) |
129 | 128 | adantllr 719 |
. . . . . . . . . 10
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → (∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)(𝐹 “ 𝑢) ⊆ 𝑏 → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤)) |
130 | 116, 129 | mpd 15 |
. . . . . . . . 9
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) |
131 | 34 | ad2antrr 726 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝐾 ∈ Top) |
132 | | cnextcn.8 |
. . . . . . . . . . . . . . 15
⊢ (𝜑 → 𝐾 ∈ Reg) |
133 | 132 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝐾 ∈ Reg) |
134 | 133 | ad2antrr 726 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → 𝐾 ∈ Reg) |
135 | | simplr 769 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → 𝑐 ∈ 𝐾) |
136 | | simprl 771 |
. . . . . . . . . . . . 13
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐) |
137 | | regsep 22087 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ Reg ∧ 𝑐 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑐)) |
138 | 134, 135,
136, 137 | syl3anc 1372 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑐)) |
139 | | sstr 3885 |
. . . . . . . . . . . . . . . 16
⊢
((((cls‘𝐾)‘𝑏) ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤) → ((cls‘𝐾)‘𝑏) ⊆ 𝑤) |
140 | 139 | expcom 417 |
. . . . . . . . . . . . . . 15
⊢ (𝑐 ⊆ 𝑤 → (((cls‘𝐾)‘𝑏) ⊆ 𝑐 → ((cls‘𝐾)‘𝑏) ⊆ 𝑤)) |
141 | 140 | anim2d 615 |
. . . . . . . . . . . . . 14
⊢ (𝑐 ⊆ 𝑤 → (((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑐) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤))) |
142 | 141 | reximdv 3183 |
. . . . . . . . . . . . 13
⊢ (𝑐 ⊆ 𝑤 → (∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑐) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤))) |
143 | 142 | ad2antll 729 |
. . . . . . . . . . . 12
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → (∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑐) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤))) |
144 | 138, 143 | mpd 15 |
. . . . . . . . . . 11
⊢
(((((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) ∧ 𝑐 ∈ 𝐾) ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)) |
145 | | simpr 488 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) |
146 | | neii2 21861 |
. . . . . . . . . . . . 13
⊢ ((𝐾 ∈ Top ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑐 ∈ 𝐾 ({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤)) |
147 | | fvex 6689 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ V |
148 | 147 | snss 4674 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ↔ {(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐) |
149 | 148 | anbi1i 627 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤) ↔ ({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤)) |
150 | 149 | biimpri 231 |
. . . . . . . . . . . . . 14
⊢
(({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤) → ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) |
151 | 150 | reximi 3157 |
. . . . . . . . . . . . 13
⊢
(∃𝑐 ∈
𝐾 ({(((𝐽CnExt𝐾)‘𝐹)‘𝑥)} ⊆ 𝑐 ∧ 𝑐 ⊆ 𝑤) → ∃𝑐 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) |
152 | 146, 151 | syl 17 |
. . . . . . . . . . . 12
⊢ ((𝐾 ∈ Top ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑐 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) |
153 | 131, 145,
152 | syl2anc 587 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑐 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑐 ∧ 𝑐 ⊆ 𝑤)) |
154 | 144, 153 | r19.29a 3199 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)) |
155 | | anass 472 |
. . . . . . . . . . . 12
⊢ (((𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) ↔ (𝑏 ∈ 𝐾 ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤))) |
156 | | opnneip 21872 |
. . . . . . . . . . . . . 14
⊢ ((𝐾 ∈ Top ∧ 𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) |
157 | 156 | 3expib 1123 |
. . . . . . . . . . . . 13
⊢ (𝐾 ∈ Top → ((𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) → 𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}))) |
158 | 157 | anim1d 614 |
. . . . . . . . . . . 12
⊢ (𝐾 ∈ Top → (((𝑏 ∈ 𝐾 ∧ (((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → (𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤))) |
159 | 155, 158 | syl5bir 246 |
. . . . . . . . . . 11
⊢ (𝐾 ∈ Top → ((𝑏 ∈ 𝐾 ∧ ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤)) → (𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)}) ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤))) |
160 | 159 | reximdv2 3181 |
. . . . . . . . . 10
⊢ (𝐾 ∈ Top → (∃𝑏 ∈ 𝐾 ((((𝐽CnExt𝐾)‘𝐹)‘𝑥) ∈ 𝑏 ∧ ((cls‘𝐾)‘𝑏) ⊆ 𝑤) → ∃𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})((cls‘𝐾)‘𝑏) ⊆ 𝑤)) |
161 | 131, 154,
160 | sylc 65 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑏 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})((cls‘𝐾)‘𝑏) ⊆ 𝑤) |
162 | 130, 161 | r19.29a 3199 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑢 ∈ (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)((cls‘𝐾)‘(𝐹 “ 𝑢)) ⊆ 𝑤) |
163 | 75, 162 | r19.29a 3199 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑑 ∈ ((nei‘𝐽)‘{𝑥})((cls‘𝐾)‘(𝐹 “ (𝑑 ∩ 𝐴))) ⊆ 𝑤) |
164 | 55, 163 | r19.29a 3199 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤)) |
165 | | simplr 769 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) |
166 | | simpll 767 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝜑) |
167 | 4 | ad2antrr 726 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝐽 ∈ Top) |
168 | | simplr 769 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ 𝐽) |
169 | 65 | eltopss 21660 |
. . . . . . . . . . . . . . 15
⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽) → 𝑣 ⊆ 𝐶) |
170 | 167, 168,
169 | syl2anc 587 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ⊆ 𝐶) |
171 | | simpr 488 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝑣) |
172 | 170, 171 | sseldd 3878 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑧 ∈ 𝐶) |
173 | | fvexd 6691 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → ((nei‘𝐽)‘{𝑧}) ∈ V) |
174 | 70 | ad2antrr 726 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝐴 ∈ V) |
175 | | opnneip 21872 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ 𝐽 ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧})) |
176 | 4, 175 | syl3an1 1164 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑣 ∈ 𝐽 ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧})) |
177 | 176 | 3expa 1119 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → 𝑣 ∈ ((nei‘𝐽)‘{𝑧})) |
178 | | elrestr 16807 |
. . . . . . . . . . . . . 14
⊢
((((nei‘𝐽)‘{𝑧}) ∈ V ∧ 𝐴 ∈ V ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑧})) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) |
179 | 173, 174,
177, 178 | syl3anc 1372 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) |
180 | 65, 42, 4, 32, 36, 69, 76, 86 | cnextfvval 22818 |
. . . . . . . . . . . . . . 15
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹)) |
181 | 180 | adantr 484 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) = ∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹)) |
182 | 32 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐾 ∈ Haus) |
183 | 76 | eleq2d 2818 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → (𝑧 ∈ ((cls‘𝐽)‘𝐴) ↔ 𝑧 ∈ 𝐶)) |
184 | 183 | biimpar 481 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ ((cls‘𝐽)‘𝐴)) |
185 | 67 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐽 ∈ (TopOn‘𝐶)) |
186 | 69 | adantr 484 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐴 ⊆ 𝐶) |
187 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐶) |
188 | | trnei 22645 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝐽 ∈ (TopOn‘𝐶) ∧ 𝐴 ⊆ 𝐶 ∧ 𝑧 ∈ 𝐶) → (𝑧 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴) ∈ (Fil‘𝐴))) |
189 | 185, 186,
187, 188 | syl3anc 1372 |
. . . . . . . . . . . . . . . . . . . 20
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝑧 ∈ ((cls‘𝐽)‘𝐴) ↔ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴) ∈ (Fil‘𝐴))) |
190 | 184, 189 | mpbid 235 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (((nei‘𝐽)‘{𝑧}) ↾t 𝐴) ∈ (Fil‘𝐴)) |
191 | 36 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐹:𝐴⟶𝐵) |
192 | | eleq1w 2815 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑧 → (𝑥 ∈ 𝐶 ↔ 𝑧 ∈ 𝐶)) |
193 | 192 | anbi2d 632 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑧 → ((𝜑 ∧ 𝑥 ∈ 𝐶) ↔ (𝜑 ∧ 𝑧 ∈ 𝐶))) |
194 | | sneq 4526 |
. . . . . . . . . . . . . . . . . . . . . . . . . 26
⊢ (𝑥 = 𝑧 → {𝑥} = {𝑧}) |
195 | 194 | fveq2d 6680 |
. . . . . . . . . . . . . . . . . . . . . . . . 25
⊢ (𝑥 = 𝑧 → ((nei‘𝐽)‘{𝑥}) = ((nei‘𝐽)‘{𝑧})) |
196 | 195 | oveq1d 7187 |
. . . . . . . . . . . . . . . . . . . . . . . 24
⊢ (𝑥 = 𝑧 → (((nei‘𝐽)‘{𝑥}) ↾t 𝐴) = (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) |
197 | 196 | oveq2d 7188 |
. . . . . . . . . . . . . . . . . . . . . . 23
⊢ (𝑥 = 𝑧 → (𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴)) = (𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))) |
198 | 197 | fveq1d 6678 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢ (𝑥 = 𝑧 → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) = ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹)) |
199 | 198 | neeq1d 2993 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝑥 = 𝑧 → (((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅ ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) ≠ ∅)) |
200 | 193, 199 | imbi12d 348 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝑥 = 𝑧 → (((𝜑 ∧ 𝑥 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑥}) ↾t 𝐴))‘𝐹) ≠ ∅) ↔ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) ≠ ∅))) |
201 | 200, 86 | chvarvv 2010 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) ≠ ∅) |
202 | 42 | hausflf2 22751 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐾 ∈ Haus ∧
(((nei‘𝐽)‘{𝑧}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝐵) ∧ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) ≠ ∅) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) ≈ 1o) |
203 | 182, 190,
191, 201, 202 | syl31anc 1374 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) ≈ 1o) |
204 | | en1b 8626 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) ≈ 1o ↔ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹)}) |
205 | 203, 204 | sylib 221 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹)}) |
206 | 205 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) = {∪ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹)}) |
207 | 109 | adantr 484 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → 𝐾 ∈ (TopOn‘𝐵)) |
208 | | flfval 22743 |
. . . . . . . . . . . . . . . . . . 19
⊢ ((𝐾 ∈ (TopOn‘𝐵) ∧ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴) ∈ (Fil‘𝐴) ∧ 𝐹:𝐴⟶𝐵) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) = (𝐾 fLim ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾t 𝐴)))) |
209 | 207, 190,
191, 208 | syl3anc 1372 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) = (𝐾 fLim ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾t 𝐴)))) |
210 | 209 | adantr 484 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) = (𝐾 fLim ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾t 𝐴)))) |
211 | 32 | uniexd 7488 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ (𝜑 → ∪ 𝐾
∈ V) |
212 | 211 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → ∪ 𝐾 ∈ V) |
213 | 42, 212 | eqeltrid 2837 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → 𝐵 ∈ V) |
214 | 190 | adantr 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → (((nei‘𝐽)‘{𝑧}) ↾t 𝐴) ∈ (Fil‘𝐴)) |
215 | | filfbas 22601 |
. . . . . . . . . . . . . . . . . . . 20
⊢
((((nei‘𝐽)‘{𝑧}) ↾t 𝐴) ∈ (Fil‘𝐴) → (((nei‘𝐽)‘{𝑧}) ↾t 𝐴) ∈ (fBas‘𝐴)) |
216 | 214, 215 | syl 17 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → (((nei‘𝐽)‘{𝑧}) ↾t 𝐴) ∈ (fBas‘𝐴)) |
217 | 36 | ad2antrr 726 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → 𝐹:𝐴⟶𝐵) |
218 | | simpr 488 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) |
219 | | fgfil 22628 |
. . . . . . . . . . . . . . . . . . . . . 22
⊢
((((nei‘𝐽)‘{𝑧}) ↾t 𝐴) ∈ (Fil‘𝐴) → (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) = (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) |
220 | 190, 219 | syl 17 |
. . . . . . . . . . . . . . . . . . . . 21
⊢ ((𝜑 ∧ 𝑧 ∈ 𝐶) → (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) = (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) |
221 | 220 | adantr 484 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) = (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) |
222 | 218, 221 | eleqtrrd 2836 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → (𝑣 ∩ 𝐴) ∈ (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾t 𝐴))) |
223 | | eqid 2738 |
. . . . . . . . . . . . . . . . . . . 20
⊢ (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) = (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) |
224 | 223 | imaelfm 22704 |
. . . . . . . . . . . . . . . . . . 19
⊢ (((𝐵 ∈ V ∧
(((nei‘𝐽)‘{𝑧}) ↾t 𝐴) ∈ (fBas‘𝐴) ∧ 𝐹:𝐴⟶𝐵) ∧ (𝑣 ∩ 𝐴) ∈ (𝐴filGen(((nei‘𝐽)‘{𝑧}) ↾t 𝐴))) → (𝐹 “ (𝑣 ∩ 𝐴)) ∈ ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾t 𝐴))) |
225 | 213, 216,
217, 222, 224 | syl31anc 1374 |
. . . . . . . . . . . . . . . . . 18
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → (𝐹 “ (𝑣 ∩ 𝐴)) ∈ ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾t 𝐴))) |
226 | | flimclsi 22731 |
. . . . . . . . . . . . . . . . . 18
⊢ ((𝐹 “ (𝑣 ∩ 𝐴)) ∈ ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → (𝐾 fLim ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾t 𝐴))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴)))) |
227 | 225, 226 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → (𝐾 fLim ((𝐵 FilMap 𝐹)‘(((nei‘𝐽)‘{𝑧}) ↾t 𝐴))) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴)))) |
228 | 210, 227 | eqsstrd 3915 |
. . . . . . . . . . . . . . . 16
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴)))) |
229 | 206, 228 | eqsstrrd 3916 |
. . . . . . . . . . . . . . 15
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → {∪
((𝐾 fLimf
(((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹)} ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴)))) |
230 | | fvex 6689 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐾 fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) ∈ V |
231 | 230 | uniex 7487 |
. . . . . . . . . . . . . . . 16
⊢ ∪ ((𝐾
fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) ∈ V |
232 | 231 | snss 4674 |
. . . . . . . . . . . . . . 15
⊢ (∪ ((𝐾
fLimf (((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ↔ {∪
((𝐾 fLimf
(((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹)} ⊆ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴)))) |
233 | 229, 232 | sylibr 237 |
. . . . . . . . . . . . . 14
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → ∪
((𝐾 fLimf
(((nei‘𝐽)‘{𝑧}) ↾t 𝐴))‘𝐹) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴)))) |
234 | 181, 233 | eqeltrd 2833 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ 𝑧 ∈ 𝐶) ∧ (𝑣 ∩ 𝐴) ∈ (((nei‘𝐽)‘{𝑧}) ↾t 𝐴)) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴)))) |
235 | 166, 172,
179, 234 | syl21anc 837 |
. . . . . . . . . . . 12
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴)))) |
236 | 235 | adantlr 715 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴)))) |
237 | 165, 236 | sseldd 3878 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) ∧ 𝑧 ∈ 𝑣) → (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤) |
238 | 237 | ralrimiva 3096 |
. . . . . . . . 9
⊢ (((𝜑 ∧ 𝑣 ∈ 𝐽) ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) → ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤) |
239 | 238 | expl 461 |
. . . . . . . 8
⊢ (𝜑 → ((𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) → ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤)) |
240 | 239 | reximdv 3183 |
. . . . . . 7
⊢ (𝜑 → (∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤)) |
241 | 240 | ad2antrr 726 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → (∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(𝑣 ∈ 𝐽 ∧ ((cls‘𝐾)‘(𝐹 “ (𝑣 ∩ 𝐴))) ⊆ 𝑤) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤)) |
242 | 164, 241 | mpd 15 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤) |
243 | 65, 42, 4, 32, 36, 69, 76, 86 | cnextf 22819 |
. . . . . . . . . 10
⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵) |
244 | 243 | ffund 6508 |
. . . . . . . . 9
⊢ (𝜑 → Fun ((𝐽CnExt𝐾)‘𝐹)) |
245 | 244 | adantr 484 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → Fun ((𝐽CnExt𝐾)‘𝐹)) |
246 | 65 | neii1 21859 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ 𝐶) |
247 | 4, 246 | sylan 583 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ 𝐶) |
248 | 243 | fdmd 6515 |
. . . . . . . . . 10
⊢ (𝜑 → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶) |
249 | 248 | adantr 484 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → dom ((𝐽CnExt𝐾)‘𝐹) = 𝐶) |
250 | 247, 249 | sseqtrrd 3918 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑣 ⊆ dom ((𝐽CnExt𝐾)‘𝐹)) |
251 | | funimass4 6736 |
. . . . . . . 8
⊢ ((Fun
((𝐽CnExt𝐾)‘𝐹) ∧ 𝑣 ⊆ dom ((𝐽CnExt𝐾)‘𝐹)) → ((((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤)) |
252 | 245, 250,
251 | syl2anc 587 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → ((((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤 ↔ ∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤)) |
253 | 252 | biimprd 251 |
. . . . . 6
⊢ ((𝜑 ∧ 𝑣 ∈ ((nei‘𝐽)‘{𝑥})) → (∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤 → (((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)) |
254 | 253 | reximdva 3184 |
. . . . 5
⊢ (𝜑 → (∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})∀𝑧 ∈ 𝑣 (((𝐽CnExt𝐾)‘𝐹)‘𝑧) ∈ 𝑤 → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)) |
255 | 1, 242, 254 | sylc 65 |
. . . 4
⊢ (((𝜑 ∧ 𝑥 ∈ 𝐶) ∧ 𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})) → ∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤) |
256 | 255 | ralrimiva 3096 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝐶) → ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤) |
257 | 256 | ralrimiva 3096 |
. 2
⊢ (𝜑 → ∀𝑥 ∈ 𝐶 ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤) |
258 | 65, 42 | cnnei 22035 |
. . 3
⊢ ((𝐽 ∈ Top ∧ 𝐾 ∈ Top ∧ ((𝐽CnExt𝐾)‘𝐹):𝐶⟶𝐵) → (((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝐶 ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)) |
259 | 4, 34, 243, 258 | syl3anc 1372 |
. 2
⊢ (𝜑 → (((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾) ↔ ∀𝑥 ∈ 𝐶 ∀𝑤 ∈ ((nei‘𝐾)‘{(((𝐽CnExt𝐾)‘𝐹)‘𝑥)})∃𝑣 ∈ ((nei‘𝐽)‘{𝑥})(((𝐽CnExt𝐾)‘𝐹) “ 𝑣) ⊆ 𝑤)) |
260 | 257, 259 | mpbird 260 |
1
⊢ (𝜑 → ((𝐽CnExt𝐾)‘𝐹) ∈ (𝐽 Cn 𝐾)) |