Step | Hyp | Ref
| Expression |
1 | | kqval.2 |
. . . . 5
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) |
2 | 1 | kqtopon 22478 |
. . . 4
⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹)) |
3 | 2 | adantr 484 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹)) |
4 | | topontop 21664 |
. . 3
⊢
((KQ‘𝐽) ∈
(TopOn‘ran 𝐹) →
(KQ‘𝐽) ∈
Top) |
5 | 3, 4 | syl 17 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Top) |
6 | | simplr 769 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝐽 ∈ Nrm) |
7 | 1 | kqid 22479 |
. . . . . . 7
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽))) |
8 | 7 | ad2antrr 726 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽))) |
9 | | simprl 771 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑧 ∈ (KQ‘𝐽)) |
10 | | cnima 22016 |
. . . . . 6
⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑧 ∈ (KQ‘𝐽)) → (◡𝐹 “ 𝑧) ∈ 𝐽) |
11 | 8, 9, 10 | syl2anc 587 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (◡𝐹 “ 𝑧) ∈ 𝐽) |
12 | | simprr 773 |
. . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧)) |
13 | 12 | elin1d 4088 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ (Clsd‘(KQ‘𝐽))) |
14 | | cnclima 22019 |
. . . . . 6
⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑤 ∈ (Clsd‘(KQ‘𝐽))) → (◡𝐹 “ 𝑤) ∈ (Clsd‘𝐽)) |
15 | 8, 13, 14 | syl2anc 587 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (◡𝐹 “ 𝑤) ∈ (Clsd‘𝐽)) |
16 | 12 | elin2d 4089 |
. . . . . 6
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → 𝑤 ∈ 𝒫 𝑧) |
17 | | elpwi 4497 |
. . . . . 6
⊢ (𝑤 ∈ 𝒫 𝑧 → 𝑤 ⊆ 𝑧) |
18 | | imass2 5939 |
. . . . . 6
⊢ (𝑤 ⊆ 𝑧 → (◡𝐹 “ 𝑤) ⊆ (◡𝐹 “ 𝑧)) |
19 | 16, 17, 18 | 3syl 18 |
. . . . 5
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → (◡𝐹 “ 𝑤) ⊆ (◡𝐹 “ 𝑧)) |
20 | | nrmsep3 22106 |
. . . . 5
⊢ ((𝐽 ∈ Nrm ∧ ((◡𝐹 “ 𝑧) ∈ 𝐽 ∧ (◡𝐹 “ 𝑤) ∈ (Clsd‘𝐽) ∧ (◡𝐹 “ 𝑤) ⊆ (◡𝐹 “ 𝑧))) → ∃𝑢 ∈ 𝐽 ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧))) |
21 | 6, 11, 15, 19, 20 | syl13anc 1373 |
. . . 4
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → ∃𝑢 ∈ 𝐽 ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧))) |
22 | | simplll 775 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝐽 ∈ (TopOn‘𝑋)) |
23 | | simprl 771 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑢 ∈ 𝐽) |
24 | 1 | kqopn 22485 |
. . . . . 6
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑢 ∈ 𝐽) → (𝐹 “ 𝑢) ∈ (KQ‘𝐽)) |
25 | 22, 23, 24 | syl2anc 587 |
. . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → (𝐹 “ 𝑢) ∈ (KQ‘𝐽)) |
26 | | simprrl 781 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → (◡𝐹 “ 𝑤) ⊆ 𝑢) |
27 | 1 | kqffn 22476 |
. . . . . . . 8
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋) |
28 | | fnfun 6438 |
. . . . . . . 8
⊢ (𝐹 Fn 𝑋 → Fun 𝐹) |
29 | 22, 27, 28 | 3syl 18 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → Fun 𝐹) |
30 | 13 | adantr 484 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑤 ∈ (Clsd‘(KQ‘𝐽))) |
31 | | eqid 2738 |
. . . . . . . . . 10
⊢ ∪ (KQ‘𝐽) = ∪
(KQ‘𝐽) |
32 | 31 | cldss 21780 |
. . . . . . . . 9
⊢ (𝑤 ∈
(Clsd‘(KQ‘𝐽))
→ 𝑤 ⊆ ∪ (KQ‘𝐽)) |
33 | 30, 32 | syl 17 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑤 ⊆ ∪
(KQ‘𝐽)) |
34 | | toponuni 21665 |
. . . . . . . . 9
⊢
((KQ‘𝐽) ∈
(TopOn‘ran 𝐹) →
ran 𝐹 = ∪ (KQ‘𝐽)) |
35 | 22, 2, 34 | 3syl 18 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ran 𝐹 = ∪
(KQ‘𝐽)) |
36 | 33, 35 | sseqtrrd 3918 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑤 ⊆ ran 𝐹) |
37 | | funimass1 6421 |
. . . . . . 7
⊢ ((Fun
𝐹 ∧ 𝑤 ⊆ ran 𝐹) → ((◡𝐹 “ 𝑤) ⊆ 𝑢 → 𝑤 ⊆ (𝐹 “ 𝑢))) |
38 | 29, 36, 37 | syl2anc 587 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((◡𝐹 “ 𝑤) ⊆ 𝑢 → 𝑤 ⊆ (𝐹 “ 𝑢))) |
39 | 26, 38 | mpd 15 |
. . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑤 ⊆ (𝐹 “ 𝑢)) |
40 | | topontop 21664 |
. . . . . . . . . 10
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) |
41 | 22, 40 | syl 17 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝐽 ∈ Top) |
42 | | elssuni 4828 |
. . . . . . . . . 10
⊢ (𝑢 ∈ 𝐽 → 𝑢 ⊆ ∪ 𝐽) |
43 | 42 | ad2antrl 728 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑢 ⊆ ∪ 𝐽) |
44 | | eqid 2738 |
. . . . . . . . . 10
⊢ ∪ 𝐽 =
∪ 𝐽 |
45 | 44 | clscld 21798 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽)
→ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) |
46 | 41, 43, 45 | syl2anc 587 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) |
47 | 1 | kqcld 22486 |
. . . . . . . 8
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑢) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽))) |
48 | 22, 46, 47 | syl2anc 587 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽))) |
49 | 44 | sscls 21807 |
. . . . . . . . 9
⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽)
→ 𝑢 ⊆
((cls‘𝐽)‘𝑢)) |
50 | 41, 43, 49 | syl2anc 587 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑢 ⊆ ((cls‘𝐽)‘𝑢)) |
51 | | imass2 5939 |
. . . . . . . 8
⊢ (𝑢 ⊆ ((cls‘𝐽)‘𝑢) → (𝐹 “ 𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) |
52 | 50, 51 | syl 17 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → (𝐹 “ 𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) |
53 | 31 | clsss2 21823 |
. . . . . . 7
⊢ (((𝐹 “ ((cls‘𝐽)‘𝑢)) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹 “ 𝑢) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) |
54 | 48, 52, 53 | syl2anc 587 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑢))) |
55 | | simprrr 782 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)) |
56 | 44 | clsss3 21810 |
. . . . . . . . . 10
⊢ ((𝐽 ∈ Top ∧ 𝑢 ⊆ ∪ 𝐽)
→ ((cls‘𝐽)‘𝑢) ⊆ ∪ 𝐽) |
57 | 41, 43, 56 | syl2anc 587 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((cls‘𝐽)‘𝑢) ⊆ ∪ 𝐽) |
58 | | fndm 6440 |
. . . . . . . . . . 11
⊢ (𝐹 Fn 𝑋 → dom 𝐹 = 𝑋) |
59 | 22, 27, 58 | 3syl 18 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → dom 𝐹 = 𝑋) |
60 | | toponuni 21665 |
. . . . . . . . . . 11
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = ∪ 𝐽) |
61 | 22, 60 | syl 17 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → 𝑋 = ∪ 𝐽) |
62 | 59, 61 | eqtrd 2773 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → dom 𝐹 = ∪ 𝐽) |
63 | 57, 62 | sseqtrrd 3918 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((cls‘𝐽)‘𝑢) ⊆ dom 𝐹) |
64 | | funimass3 6831 |
. . . . . . . 8
⊢ ((Fun
𝐹 ∧ ((cls‘𝐽)‘𝑢) ⊆ dom 𝐹) → ((𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧 ↔ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧))) |
65 | 29, 63, 64 | syl2anc 587 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧 ↔ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧))) |
66 | 55, 65 | mpbird 260 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → (𝐹 “ ((cls‘𝐽)‘𝑢)) ⊆ 𝑧) |
67 | 54, 66 | sstrd 3887 |
. . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧) |
68 | | sseq2 3903 |
. . . . . . 7
⊢ (𝑚 = (𝐹 “ 𝑢) → (𝑤 ⊆ 𝑚 ↔ 𝑤 ⊆ (𝐹 “ 𝑢))) |
69 | | fveq2 6674 |
. . . . . . . 8
⊢ (𝑚 = (𝐹 “ 𝑢) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢))) |
70 | 69 | sseq1d 3908 |
. . . . . . 7
⊢ (𝑚 = (𝐹 “ 𝑢) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧 ↔ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧)) |
71 | 68, 70 | anbi12d 634 |
. . . . . 6
⊢ (𝑚 = (𝐹 “ 𝑢) → ((𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧) ↔ (𝑤 ⊆ (𝐹 “ 𝑢) ∧ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧))) |
72 | 71 | rspcev 3526 |
. . . . 5
⊢ (((𝐹 “ 𝑢) ∈ (KQ‘𝐽) ∧ (𝑤 ⊆ (𝐹 “ 𝑢) ∧ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑢)) ⊆ 𝑧)) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧)) |
73 | 25, 39, 67, 72 | syl12anc 836 |
. . . 4
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) ∧ (𝑢 ∈ 𝐽 ∧ ((◡𝐹 “ 𝑤) ⊆ 𝑢 ∧ ((cls‘𝐽)‘𝑢) ⊆ (◡𝐹 “ 𝑧)))) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧)) |
74 | 21, 73 | rexlimddv 3201 |
. . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) ∧ (𝑧 ∈ (KQ‘𝐽) ∧ 𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧))) → ∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧)) |
75 | 74 | ralrimivva 3103 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → ∀𝑧 ∈ (KQ‘𝐽)∀𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧)∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧)) |
76 | | isnrm 22086 |
. 2
⊢
((KQ‘𝐽) ∈
Nrm ↔ ((KQ‘𝐽)
∈ Top ∧ ∀𝑧
∈ (KQ‘𝐽)∀𝑤 ∈ ((Clsd‘(KQ‘𝐽)) ∩ 𝒫 𝑧)∃𝑚 ∈ (KQ‘𝐽)(𝑤 ⊆ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑧))) |
77 | 5, 75, 76 | sylanbrc 586 |
1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Nrm) → (KQ‘𝐽) ∈ Nrm) |