Step | Hyp | Ref
| Expression |
1 | | llycmpkgen2.2 |
. 2
⊢ (𝜑 → 𝐽 ∈ Top) |
2 | | elssuni 4467 |
. . . . . . . . . . 11
⊢ (𝑢 ∈
(𝑘Gen‘𝐽)
→ 𝑢 ⊆ ∪ (𝑘Gen‘𝐽)) |
3 | 2 | adantl 482 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑢 ⊆ ∪
(𝑘Gen‘𝐽)) |
4 | | iskgen3.1 |
. . . . . . . . . . . . 13
⊢ 𝑋 = ∪
𝐽 |
5 | 4 | kgenuni 21342 |
. . . . . . . . . . . 12
⊢ (𝐽 ∈ Top → 𝑋 = ∪
(𝑘Gen‘𝐽)) |
6 | 1, 5 | syl 17 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝑋 = ∪
(𝑘Gen‘𝐽)) |
7 | 6 | adantr 481 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑋 = ∪
(𝑘Gen‘𝐽)) |
8 | 3, 7 | sseqtr4d 3642 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) → 𝑢 ⊆ 𝑋) |
9 | 8 | sselda 3603 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) → 𝑥 ∈ 𝑋) |
10 | | llycmpkgen2.3 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp) |
11 | 10 | adantlr 751 |
. . . . . . . 8
⊢ (((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑋) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp) |
12 | 9, 11 | syldan 487 |
. . . . . . 7
⊢ (((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) → ∃𝑘 ∈ ((nei‘𝐽)‘{𝑥})(𝐽 ↾t 𝑘) ∈ Comp) |
13 | 1 | ad3antrrr 766 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝐽 ∈ Top) |
14 | | difss 3737 |
. . . . . . . . . 10
⊢ (𝑋 ∖ (𝑘 ∖ 𝑢)) ⊆ 𝑋 |
15 | 4 | ntropn 20853 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ (𝑘 ∖ 𝑢)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∈ 𝐽) |
16 | 13, 14, 15 | sylancl 694 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∈ 𝐽) |
17 | | simprl 794 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑘 ∈ ((nei‘𝐽)‘{𝑥})) |
18 | 4 | neii1 20910 |
. . . . . . . . . . 11
⊢ ((𝐽 ∈ Top ∧ 𝑘 ∈ ((nei‘𝐽)‘{𝑥})) → 𝑘 ⊆ 𝑋) |
19 | 13, 17, 18 | syl2anc 693 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑘 ⊆ 𝑋) |
20 | 4 | ntropn 20853 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝑘 ⊆ 𝑋) → ((int‘𝐽)‘𝑘) ∈ 𝐽) |
21 | 13, 19, 20 | syl2anc 693 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘𝐽)‘𝑘) ∈ 𝐽) |
22 | | inopn 20704 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧
((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∈ 𝐽 ∧ ((int‘𝐽)‘𝑘) ∈ 𝐽) → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽) |
23 | 13, 16, 21, 22 | syl3anc 1326 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽) |
24 | | inss1 3833 |
. . . . . . . . . . 11
⊢
(((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘))) ∩ 𝑘) ⊆ ((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘))) |
25 | | simplr 792 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ 𝑢) |
26 | 4 | ntrss2 20861 |
. . . . . . . . . . . . . . 15
⊢ ((𝐽 ∈ Top ∧ 𝑘 ⊆ 𝑋) → ((int‘𝐽)‘𝑘) ⊆ 𝑘) |
27 | 13, 19, 26 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘𝐽)‘𝑘) ⊆ 𝑘) |
28 | 9 | adantr 481 |
. . . . . . . . . . . . . . . . . 18
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ 𝑋) |
29 | 28 | snssd 4340 |
. . . . . . . . . . . . . . . . 17
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → {𝑥} ⊆ 𝑋) |
30 | 4 | neiint 20908 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐽 ∈ Top ∧ {𝑥} ⊆ 𝑋 ∧ 𝑘 ⊆ 𝑋) → (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘))) |
31 | 13, 29, 19, 30 | syl3anc 1326 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘))) |
32 | 17, 31 | mpbid 222 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → {𝑥} ⊆ ((int‘𝐽)‘𝑘)) |
33 | | vex 3203 |
. . . . . . . . . . . . . . . 16
⊢ 𝑥 ∈ V |
34 | 33 | snss 4316 |
. . . . . . . . . . . . . . 15
⊢ (𝑥 ∈ ((int‘𝐽)‘𝑘) ↔ {𝑥} ⊆ ((int‘𝐽)‘𝑘)) |
35 | 32, 34 | sylibr 224 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘𝑘)) |
36 | 27, 35 | sseldd 3604 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ 𝑘) |
37 | 25, 36 | elind 3798 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ (𝑢 ∩ 𝑘)) |
38 | | simpllr 799 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑢 ∈ (𝑘Gen‘𝐽)) |
39 | | simprr 796 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ Comp) |
40 | | kgeni 21340 |
. . . . . . . . . . . . . . 15
⊢ ((𝑢 ∈
(𝑘Gen‘𝐽)
∧ (𝐽
↾t 𝑘)
∈ Comp) → (𝑢
∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) |
41 | 38, 39, 40 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑢 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘)) |
42 | | vex 3203 |
. . . . . . . . . . . . . . . 16
⊢ 𝑘 ∈ V |
43 | | resttop 20964 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐽 ∈ Top ∧ 𝑘 ∈ V) → (𝐽 ↾t 𝑘) ∈ Top) |
44 | 13, 42, 43 | sylancl 694 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝐽 ↾t 𝑘) ∈ Top) |
45 | | inss2 3834 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 ∩ 𝑘) ⊆ 𝑘 |
46 | 4 | restuni 20966 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐽 ∈ Top ∧ 𝑘 ⊆ 𝑋) → 𝑘 = ∪ (𝐽 ↾t 𝑘)) |
47 | 13, 19, 46 | syl2anc 693 |
. . . . . . . . . . . . . . . 16
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑘 = ∪ (𝐽 ↾t 𝑘)) |
48 | 45, 47 | syl5sseq 3653 |
. . . . . . . . . . . . . . 15
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑢 ∩ 𝑘) ⊆ ∪ (𝐽 ↾t 𝑘)) |
49 | | eqid 2622 |
. . . . . . . . . . . . . . . 16
⊢ ∪ (𝐽
↾t 𝑘) =
∪ (𝐽 ↾t 𝑘) |
50 | 49 | isopn3 20870 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ↾t 𝑘) ∈ Top ∧ (𝑢 ∩ 𝑘) ⊆ ∪ (𝐽 ↾t 𝑘)) → ((𝑢 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘) ↔ ((int‘(𝐽 ↾t 𝑘))‘(𝑢 ∩ 𝑘)) = (𝑢 ∩ 𝑘))) |
51 | 44, 48, 50 | syl2anc 693 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((𝑢 ∩ 𝑘) ∈ (𝐽 ↾t 𝑘) ↔ ((int‘(𝐽 ↾t 𝑘))‘(𝑢 ∩ 𝑘)) = (𝑢 ∩ 𝑘))) |
52 | 41, 51 | mpbid 222 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘(𝐽 ↾t 𝑘))‘(𝑢 ∩ 𝑘)) = (𝑢 ∩ 𝑘)) |
53 | 45 | a1i 11 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑢 ∩ 𝑘) ⊆ 𝑘) |
54 | | eqid 2622 |
. . . . . . . . . . . . . . 15
⊢ (𝐽 ↾t 𝑘) = (𝐽 ↾t 𝑘) |
55 | 4, 54 | restntr 20986 |
. . . . . . . . . . . . . 14
⊢ ((𝐽 ∈ Top ∧ 𝑘 ⊆ 𝑋 ∧ (𝑢 ∩ 𝑘) ⊆ 𝑘) → ((int‘(𝐽 ↾t 𝑘))‘(𝑢 ∩ 𝑘)) = (((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘))) ∩ 𝑘)) |
56 | 13, 19, 53, 55 | syl3anc 1326 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘(𝐽 ↾t 𝑘))‘(𝑢 ∩ 𝑘)) = (((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘))) ∩ 𝑘)) |
57 | 52, 56 | eqtr3d 2658 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑢 ∩ 𝑘) = (((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘))) ∩ 𝑘)) |
58 | 37, 57 | eleqtrd 2703 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ (((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘))) ∩ 𝑘)) |
59 | 24, 58 | sseldi 3601 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘)))) |
60 | | undif3 3888 |
. . . . . . . . . . . . 13
⊢ ((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘)) = (((𝑢 ∩ 𝑘) ∪ 𝑋) ∖ (𝑘 ∖ (𝑢 ∩ 𝑘))) |
61 | | incom 3805 |
. . . . . . . . . . . . . . . 16
⊢ (𝑢 ∩ 𝑘) = (𝑘 ∩ 𝑢) |
62 | 61 | difeq2i 3725 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∖ (𝑢 ∩ 𝑘)) = (𝑘 ∖ (𝑘 ∩ 𝑢)) |
63 | | difin 3861 |
. . . . . . . . . . . . . . 15
⊢ (𝑘 ∖ (𝑘 ∩ 𝑢)) = (𝑘 ∖ 𝑢) |
64 | 62, 63 | eqtri 2644 |
. . . . . . . . . . . . . 14
⊢ (𝑘 ∖ (𝑢 ∩ 𝑘)) = (𝑘 ∖ 𝑢) |
65 | 64 | difeq2i 3725 |
. . . . . . . . . . . . 13
⊢ (((𝑢 ∩ 𝑘) ∪ 𝑋) ∖ (𝑘 ∖ (𝑢 ∩ 𝑘))) = (((𝑢 ∩ 𝑘) ∪ 𝑋) ∖ (𝑘 ∖ 𝑢)) |
66 | 60, 65 | eqtri 2644 |
. . . . . . . . . . . 12
⊢ ((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘)) = (((𝑢 ∩ 𝑘) ∪ 𝑋) ∖ (𝑘 ∖ 𝑢)) |
67 | 45, 19 | syl5ss 3614 |
. . . . . . . . . . . . . 14
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (𝑢 ∩ 𝑘) ⊆ 𝑋) |
68 | | ssequn1 3783 |
. . . . . . . . . . . . . 14
⊢ ((𝑢 ∩ 𝑘) ⊆ 𝑋 ↔ ((𝑢 ∩ 𝑘) ∪ 𝑋) = 𝑋) |
69 | 67, 68 | sylib 208 |
. . . . . . . . . . . . 13
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((𝑢 ∩ 𝑘) ∪ 𝑋) = 𝑋) |
70 | 69 | difeq1d 3727 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (((𝑢 ∩ 𝑘) ∪ 𝑋) ∖ (𝑘 ∖ 𝑢)) = (𝑋 ∖ (𝑘 ∖ 𝑢))) |
71 | 66, 70 | syl5eq 2668 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘)) = (𝑋 ∖ (𝑘 ∖ 𝑢))) |
72 | 71 | fveq2d 6195 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘𝐽)‘((𝑢 ∩ 𝑘) ∪ (𝑋 ∖ 𝑘))) = ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢)))) |
73 | 59, 72 | eleqtrd 2703 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢)))) |
74 | 73, 35 | elind 3798 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → 𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘))) |
75 | | sslin 3839 |
. . . . . . . . . 10
⊢
(((int‘𝐽)‘𝑘) ⊆ 𝑘 → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ 𝑘)) |
76 | 27, 75 | syl 17 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ 𝑘)) |
77 | 4 | ntrss2 20861 |
. . . . . . . . . . . 12
⊢ ((𝐽 ∈ Top ∧ (𝑋 ∖ (𝑘 ∖ 𝑢)) ⊆ 𝑋) → ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ⊆ (𝑋 ∖ (𝑘 ∖ 𝑢))) |
78 | 13, 14, 77 | sylancl 694 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ⊆ (𝑋 ∖ (𝑘 ∖ 𝑢))) |
79 | 78 | difss2d 3740 |
. . . . . . . . . . . 12
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ⊆ 𝑋) |
80 | | reldisj 4020 |
. . . . . . . . . . . 12
⊢
(((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ⊆ 𝑋 → ((((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ (𝑘 ∖ 𝑢)) = ∅ ↔ ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ⊆ (𝑋 ∖ (𝑘 ∖ 𝑢)))) |
81 | 79, 80 | syl 17 |
. . . . . . . . . . 11
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ((((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ (𝑘 ∖ 𝑢)) = ∅ ↔ ((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ⊆ (𝑋 ∖ (𝑘 ∖ 𝑢)))) |
82 | 78, 81 | mpbird 247 |
. . . . . . . . . 10
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ (𝑘 ∖ 𝑢)) = ∅) |
83 | | inssdif0 3947 |
. . . . . . . . . 10
⊢
((((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ 𝑘) ⊆ 𝑢 ↔ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ (𝑘 ∖ 𝑢)) = ∅) |
84 | 82, 83 | sylibr 224 |
. . . . . . . . 9
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ 𝑘) ⊆ 𝑢) |
85 | 76, 84 | sstrd 3613 |
. . . . . . . 8
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢) |
86 | | eleq2 2690 |
. . . . . . . . . 10
⊢ (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) → (𝑥 ∈ 𝑧 ↔ 𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)))) |
87 | | sseq1 3626 |
. . . . . . . . . 10
⊢ (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) → (𝑧 ⊆ 𝑢 ↔ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢)) |
88 | 86, 87 | anbi12d 747 |
. . . . . . . . 9
⊢ (𝑧 = (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) → ((𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢) ↔ (𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∧ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢))) |
89 | 88 | rspcev 3309 |
. . . . . . . 8
⊢
(((((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∈ 𝐽 ∧ (𝑥 ∈ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ∧ (((int‘𝐽)‘(𝑋 ∖ (𝑘 ∖ 𝑢))) ∩ ((int‘𝐽)‘𝑘)) ⊆ 𝑢)) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢)) |
90 | 23, 74, 85, 89 | syl12anc 1324 |
. . . . . . 7
⊢ ((((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) ∧ (𝑘 ∈ ((nei‘𝐽)‘{𝑥}) ∧ (𝐽 ↾t 𝑘) ∈ Comp)) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢)) |
91 | 12, 90 | rexlimddv 3035 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) ∧ 𝑥 ∈ 𝑢) → ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢)) |
92 | 91 | ralrimiva 2966 |
. . . . 5
⊢ ((𝜑 ∧ 𝑢 ∈ (𝑘Gen‘𝐽)) → ∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢)) |
93 | 92 | ex 450 |
. . . 4
⊢ (𝜑 → (𝑢 ∈ (𝑘Gen‘𝐽) → ∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢))) |
94 | | eltop2 20779 |
. . . . 5
⊢ (𝐽 ∈ Top → (𝑢 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢))) |
95 | 1, 94 | syl 17 |
. . . 4
⊢ (𝜑 → (𝑢 ∈ 𝐽 ↔ ∀𝑥 ∈ 𝑢 ∃𝑧 ∈ 𝐽 (𝑥 ∈ 𝑧 ∧ 𝑧 ⊆ 𝑢))) |
96 | 93, 95 | sylibrd 249 |
. . 3
⊢ (𝜑 → (𝑢 ∈ (𝑘Gen‘𝐽) → 𝑢 ∈ 𝐽)) |
97 | 96 | ssrdv 3609 |
. 2
⊢ (𝜑 → (𝑘Gen‘𝐽) ⊆ 𝐽) |
98 | | iskgen2 21351 |
. 2
⊢ (𝐽 ∈ ran 𝑘Gen ↔
(𝐽 ∈ Top ∧
(𝑘Gen‘𝐽)
⊆ 𝐽)) |
99 | 1, 97, 98 | sylanbrc 698 |
1
⊢ (𝜑 → 𝐽 ∈ ran 𝑘Gen) |