Step | Hyp | Ref
| Expression |
1 | | kqval.2 |
. . . . 5
⊢ 𝐹 = (𝑥 ∈ 𝑋 ↦ {𝑦 ∈ 𝐽 ∣ 𝑥 ∈ 𝑦}) |
2 | 1 | kqtopon 22653 |
. . . 4
⊢ (𝐽 ∈ (TopOn‘𝑋) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹)) |
3 | 2 | adantr 484 |
. . 3
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ (TopOn‘ran 𝐹)) |
4 | | topontop 21839 |
. . 3
⊢
((KQ‘𝐽) ∈
(TopOn‘ran 𝐹) →
(KQ‘𝐽) ∈
Top) |
5 | 3, 4 | syl 17 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Top) |
6 | | toponss 21853 |
. . . . . . . 8
⊢
(((KQ‘𝐽)
∈ (TopOn‘ran 𝐹)
∧ 𝑎 ∈
(KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹) |
7 | 3, 6 | sylan 583 |
. . . . . . 7
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝑎 ⊆ ran 𝐹) |
8 | 7 | sselda 3917 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → 𝑏 ∈ ran 𝐹) |
9 | 1 | kqffn 22651 |
. . . . . . . 8
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 Fn 𝑋) |
10 | 9 | ad3antrrr 730 |
. . . . . . 7
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → 𝐹 Fn 𝑋) |
11 | | fvelrnb 6794 |
. . . . . . 7
⊢ (𝐹 Fn 𝑋 → (𝑏 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑏)) |
12 | 10, 11 | syl 17 |
. . . . . 6
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → (𝑏 ∈ ran 𝐹 ↔ ∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑏)) |
13 | 8, 12 | mpbid 235 |
. . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → ∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑏) |
14 | | simpllr 776 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝐽 ∈ Reg) |
15 | 1 | kqid 22654 |
. . . . . . . . . . . . . . 15
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽))) |
16 | 15 | ad3antrrr 730 |
. . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝐹 ∈ (𝐽 Cn (KQ‘𝐽))) |
17 | | simplr 769 |
. . . . . . . . . . . . . 14
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝑎 ∈ (KQ‘𝐽)) |
18 | | cnima 22191 |
. . . . . . . . . . . . . 14
⊢ ((𝐹 ∈ (𝐽 Cn (KQ‘𝐽)) ∧ 𝑎 ∈ (KQ‘𝐽)) → (◡𝐹 “ 𝑎) ∈ 𝐽) |
19 | 16, 17, 18 | syl2anc 587 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → (◡𝐹 “ 𝑎) ∈ 𝐽) |
20 | 9 | adantr 484 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → 𝐹 Fn 𝑋) |
21 | 20 | adantr 484 |
. . . . . . . . . . . . . . 15
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → 𝐹 Fn 𝑋) |
22 | | elpreima 6899 |
. . . . . . . . . . . . . . 15
⊢ (𝐹 Fn 𝑋 → (𝑧 ∈ (◡𝐹 “ 𝑎) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎))) |
23 | 21, 22 | syl 17 |
. . . . . . . . . . . . . 14
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) → (𝑧 ∈ (◡𝐹 “ 𝑎) ↔ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎))) |
24 | 23 | biimpar 481 |
. . . . . . . . . . . . 13
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → 𝑧 ∈ (◡𝐹 “ 𝑎)) |
25 | | regsep 22260 |
. . . . . . . . . . . . 13
⊢ ((𝐽 ∈ Reg ∧ (◡𝐹 “ 𝑎) ∈ 𝐽 ∧ 𝑧 ∈ (◡𝐹 “ 𝑎)) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎))) |
26 | 14, 19, 24, 25 | syl3anc 1373 |
. . . . . . . . . . . 12
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → ∃𝑤 ∈ 𝐽 (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎))) |
27 | | simp-4l 783 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → 𝐽 ∈ (TopOn‘𝑋)) |
28 | | simprl 771 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → 𝑤 ∈ 𝐽) |
29 | 1 | kqopn 22660 |
. . . . . . . . . . . . . 14
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽)) |
30 | 27, 28, 29 | syl2anc 587 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → (𝐹 “ 𝑤) ∈ (KQ‘𝐽)) |
31 | | simprrl 781 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → 𝑧 ∈ 𝑤) |
32 | | simplrl 777 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → 𝑧 ∈ 𝑋) |
33 | 1 | kqfvima 22656 |
. . . . . . . . . . . . . . 15
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝑤 ∈ 𝐽 ∧ 𝑧 ∈ 𝑋) → (𝑧 ∈ 𝑤 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑤))) |
34 | 27, 28, 32, 33 | syl3anc 1373 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → (𝑧 ∈ 𝑤 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑤))) |
35 | 31, 34 | mpbid 235 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → (𝐹‘𝑧) ∈ (𝐹 “ 𝑤)) |
36 | | topontop 21839 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝐽 ∈ (TopOn‘𝑋) → 𝐽 ∈ Top) |
37 | 27, 36 | syl 17 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → 𝐽 ∈ Top) |
38 | | elssuni 4867 |
. . . . . . . . . . . . . . . . . 18
⊢ (𝑤 ∈ 𝐽 → 𝑤 ⊆ ∪ 𝐽) |
39 | 38 | ad2antrl 728 |
. . . . . . . . . . . . . . . . 17
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → 𝑤 ⊆ ∪ 𝐽) |
40 | | eqid 2739 |
. . . . . . . . . . . . . . . . . 18
⊢ ∪ 𝐽 =
∪ 𝐽 |
41 | 40 | clscld 21973 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽)
→ ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽)) |
42 | 37, 39, 41 | syl2anc 587 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽)) |
43 | 1 | kqcld 22661 |
. . . . . . . . . . . . . . . 16
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ ((cls‘𝐽)‘𝑤) ∈ (Clsd‘𝐽)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽))) |
44 | 27, 42, 43 | syl2anc 587 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽))) |
45 | 40 | sscls 21982 |
. . . . . . . . . . . . . . . . 17
⊢ ((𝐽 ∈ Top ∧ 𝑤 ⊆ ∪ 𝐽)
→ 𝑤 ⊆
((cls‘𝐽)‘𝑤)) |
46 | 37, 39, 45 | syl2anc 587 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → 𝑤 ⊆ ((cls‘𝐽)‘𝑤)) |
47 | | imass2 5987 |
. . . . . . . . . . . . . . . 16
⊢ (𝑤 ⊆ ((cls‘𝐽)‘𝑤) → (𝐹 “ 𝑤) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤))) |
48 | 46, 47 | syl 17 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → (𝐹 “ 𝑤) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤))) |
49 | | eqid 2739 |
. . . . . . . . . . . . . . . 16
⊢ ∪ (KQ‘𝐽) = ∪
(KQ‘𝐽) |
50 | 49 | clsss2 21998 |
. . . . . . . . . . . . . . 15
⊢ (((𝐹 “ ((cls‘𝐽)‘𝑤)) ∈ (Clsd‘(KQ‘𝐽)) ∧ (𝐹 “ 𝑤) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤))) |
51 | 44, 48, 50 | syl2anc 587 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ (𝐹 “ ((cls‘𝐽)‘𝑤))) |
52 | 20 | ad3antrrr 730 |
. . . . . . . . . . . . . . . 16
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → 𝐹 Fn 𝑋) |
53 | | fnfun 6499 |
. . . . . . . . . . . . . . . 16
⊢ (𝐹 Fn 𝑋 → Fun 𝐹) |
54 | 52, 53 | syl 17 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → Fun 𝐹) |
55 | | simprrr 782 |
. . . . . . . . . . . . . . 15
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)) |
56 | | funimass2 6483 |
. . . . . . . . . . . . . . 15
⊢ ((Fun
𝐹 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑎) |
57 | 54, 55, 56 | syl2anc 587 |
. . . . . . . . . . . . . 14
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → (𝐹 “ ((cls‘𝐽)‘𝑤)) ⊆ 𝑎) |
58 | 51, 57 | sstrd 3927 |
. . . . . . . . . . . . 13
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ 𝑎) |
59 | | eleq2 2828 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = (𝐹 “ 𝑤) → ((𝐹‘𝑧) ∈ 𝑚 ↔ (𝐹‘𝑧) ∈ (𝐹 “ 𝑤))) |
60 | | fveq2 6738 |
. . . . . . . . . . . . . . . 16
⊢ (𝑚 = (𝐹 “ 𝑤) → ((cls‘(KQ‘𝐽))‘𝑚) = ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤))) |
61 | 60 | sseq1d 3948 |
. . . . . . . . . . . . . . 15
⊢ (𝑚 = (𝐹 “ 𝑤) → (((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎 ↔ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ 𝑎)) |
62 | 59, 61 | anbi12d 634 |
. . . . . . . . . . . . . 14
⊢ (𝑚 = (𝐹 “ 𝑤) → (((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑤) ∧ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ 𝑎))) |
63 | 62 | rspcev 3551 |
. . . . . . . . . . . . 13
⊢ (((𝐹 “ 𝑤) ∈ (KQ‘𝐽) ∧ ((𝐹‘𝑧) ∈ (𝐹 “ 𝑤) ∧ ((cls‘(KQ‘𝐽))‘(𝐹 “ 𝑤)) ⊆ 𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)) |
64 | 30, 35, 58, 63 | syl12anc 837 |
. . . . . . . . . . . 12
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) ∧ (𝑤 ∈ 𝐽 ∧ (𝑧 ∈ 𝑤 ∧ ((cls‘𝐽)‘𝑤) ⊆ (◡𝐹 “ 𝑎)))) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)) |
65 | 26, 64 | rexlimddv 3219 |
. . . . . . . . . . 11
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ (𝑧 ∈ 𝑋 ∧ (𝐹‘𝑧) ∈ 𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)) |
66 | 65 | expr 460 |
. . . . . . . . . 10
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) ∈ 𝑎 → ∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))) |
67 | | eleq1 2827 |
. . . . . . . . . . 11
⊢ ((𝐹‘𝑧) = 𝑏 → ((𝐹‘𝑧) ∈ 𝑎 ↔ 𝑏 ∈ 𝑎)) |
68 | | eleq1 2827 |
. . . . . . . . . . . . 13
⊢ ((𝐹‘𝑧) = 𝑏 → ((𝐹‘𝑧) ∈ 𝑚 ↔ 𝑏 ∈ 𝑚)) |
69 | 68 | anbi1d 633 |
. . . . . . . . . . . 12
⊢ ((𝐹‘𝑧) = 𝑏 → (((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ (𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))) |
70 | 69 | rexbidv 3225 |
. . . . . . . . . . 11
⊢ ((𝐹‘𝑧) = 𝑏 → (∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎) ↔ ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))) |
71 | 67, 70 | imbi12d 348 |
. . . . . . . . . 10
⊢ ((𝐹‘𝑧) = 𝑏 → (((𝐹‘𝑧) ∈ 𝑎 → ∃𝑚 ∈ (KQ‘𝐽)((𝐹‘𝑧) ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)) ↔ (𝑏 ∈ 𝑎 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))) |
72 | 66, 71 | syl5ibcom 248 |
. . . . . . . . 9
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) = 𝑏 → (𝑏 ∈ 𝑎 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))) |
73 | 72 | com23 86 |
. . . . . . . 8
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧 ∈ 𝑋) → (𝑏 ∈ 𝑎 → ((𝐹‘𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)))) |
74 | 73 | imp 410 |
. . . . . . 7
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑧 ∈ 𝑋) ∧ 𝑏 ∈ 𝑎) → ((𝐹‘𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))) |
75 | 74 | an32s 652 |
. . . . . 6
⊢
(((((𝐽 ∈
(TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) ∧ 𝑧 ∈ 𝑋) → ((𝐹‘𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))) |
76 | 75 | rexlimdva 3212 |
. . . . 5
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → (∃𝑧 ∈ 𝑋 (𝐹‘𝑧) = 𝑏 → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))) |
77 | 13, 76 | mpd 15 |
. . . 4
⊢ ((((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ 𝑎 ∈ (KQ‘𝐽)) ∧ 𝑏 ∈ 𝑎) → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)) |
78 | 77 | anasss 470 |
. . 3
⊢ (((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) ∧ (𝑎 ∈ (KQ‘𝐽) ∧ 𝑏 ∈ 𝑎)) → ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)) |
79 | 78 | ralrimivva 3114 |
. 2
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → ∀𝑎 ∈ (KQ‘𝐽)∀𝑏 ∈ 𝑎 ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎)) |
80 | | isreg 22258 |
. 2
⊢
((KQ‘𝐽) ∈
Reg ↔ ((KQ‘𝐽)
∈ Top ∧ ∀𝑎
∈ (KQ‘𝐽)∀𝑏 ∈ 𝑎 ∃𝑚 ∈ (KQ‘𝐽)(𝑏 ∈ 𝑚 ∧ ((cls‘(KQ‘𝐽))‘𝑚) ⊆ 𝑎))) |
81 | 5, 79, 80 | sylanbrc 586 |
1
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ Reg) → (KQ‘𝐽) ∈ Reg) |