Step | Hyp | Ref
| Expression |
1 | | ptcnp.3 |
. . . . . . . . . 10
⊢ (𝜑 → 𝐽 ∈ (TopOn‘𝑋)) |
2 | 1 | adantr 480 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → 𝐽 ∈ (TopOn‘𝑋)) |
3 | | ptcnp.5 |
. . . . . . . . . . 11
⊢ (𝜑 → 𝐹:𝐼⟶Top) |
4 | 3 | ffvelrnda 6267 |
. . . . . . . . . 10
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ Top) |
5 | | eqid 2610 |
. . . . . . . . . . 11
⊢ ∪ (𝐹‘𝑘) = ∪ (𝐹‘𝑘) |
6 | 5 | toptopon 20548 |
. . . . . . . . . 10
⊢ ((𝐹‘𝑘) ∈ Top ↔ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘))) |
7 | 4, 6 | sylib 207 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘))) |
8 | | ptcnp.7 |
. . . . . . . . 9
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝐷)) |
9 | | cnpf2 20864 |
. . . . . . . . 9
⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘)) ∧ (𝑥 ∈ 𝑋 ↦ 𝐴) ∈ ((𝐽 CnP (𝐹‘𝑘))‘𝐷)) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘)) |
10 | 2, 7, 8, 9 | syl3anc 1318 |
. . . . . . . 8
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘)) |
11 | | eqid 2610 |
. . . . . . . . 9
⊢ (𝑥 ∈ 𝑋 ↦ 𝐴) = (𝑥 ∈ 𝑋 ↦ 𝐴) |
12 | 11 | fmpt 6289 |
. . . . . . . 8
⊢
(∀𝑥 ∈
𝑋 𝐴 ∈ ∪ (𝐹‘𝑘) ↔ (𝑥 ∈ 𝑋 ↦ 𝐴):𝑋⟶∪ (𝐹‘𝑘)) |
13 | 10, 12 | sylibr 223 |
. . . . . . 7
⊢ ((𝜑 ∧ 𝑘 ∈ 𝐼) → ∀𝑥 ∈ 𝑋 𝐴 ∈ ∪ (𝐹‘𝑘)) |
14 | 13 | r19.21bi 2916 |
. . . . . 6
⊢ (((𝜑 ∧ 𝑘 ∈ 𝐼) ∧ 𝑥 ∈ 𝑋) → 𝐴 ∈ ∪ (𝐹‘𝑘)) |
15 | 14 | an32s 842 |
. . . . 5
⊢ (((𝜑 ∧ 𝑥 ∈ 𝑋) ∧ 𝑘 ∈ 𝐼) → 𝐴 ∈ ∪ (𝐹‘𝑘)) |
16 | 15 | ralrimiva 2949 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘)) |
17 | | ptcnp.4 |
. . . . . 6
⊢ (𝜑 → 𝐼 ∈ 𝑉) |
18 | 17 | adantr 480 |
. . . . 5
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → 𝐼 ∈ 𝑉) |
19 | | mptelixpg 7831 |
. . . . 5
⊢ (𝐼 ∈ 𝑉 → ((𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) ↔ ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘))) |
20 | 18, 19 | syl 17 |
. . . 4
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → ((𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) ↔ ∀𝑘 ∈ 𝐼 𝐴 ∈ ∪ (𝐹‘𝑘))) |
21 | 16, 20 | mpbird 246 |
. . 3
⊢ ((𝜑 ∧ 𝑥 ∈ 𝑋) → (𝑘 ∈ 𝐼 ↦ 𝐴) ∈ X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘)) |
22 | | eqid 2610 |
. . 3
⊢ (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) = (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) |
23 | 21, 22 | fmptd 6292 |
. 2
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘)) |
24 | | df-3an 1033 |
. . . . . . . 8
⊢ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛))) |
25 | | ptcnp.2 |
. . . . . . . . . . . . 13
⊢ 𝐾 =
(∏t‘𝐹) |
26 | | ptcnp.6 |
. . . . . . . . . . . . 13
⊢ (𝜑 → 𝐷 ∈ 𝑋) |
27 | | nfv 1830 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘(𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) |
28 | | nfv 1830 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑘(𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) |
29 | | nfcv 2751 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘𝑋 |
30 | | nfmpt1 4675 |
. . . . . . . . . . . . . . . . . 18
⊢
Ⅎ𝑘(𝑘 ∈ 𝐼 ↦ 𝐴) |
31 | 29, 30 | nfmpt 4674 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘(𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) |
32 | | nfcv 2751 |
. . . . . . . . . . . . . . . . 17
⊢
Ⅎ𝑘𝐷 |
33 | 31, 32 | nffv 6110 |
. . . . . . . . . . . . . . . 16
⊢
Ⅎ𝑘((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) |
34 | 33 | nfel1 2765 |
. . . . . . . . . . . . . . 15
⊢
Ⅎ𝑘((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛) |
35 | 28, 34 | nfan 1816 |
. . . . . . . . . . . . . 14
⊢
Ⅎ𝑘((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)) |
36 | 27, 35 | nfan 1816 |
. . . . . . . . . . . . 13
⊢
Ⅎ𝑘((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛))) |
37 | | simprll 798 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → 𝑔 Fn 𝐼) |
38 | | simprlr 799 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) |
39 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → (𝑔‘𝑛) = (𝑔‘𝑘)) |
40 | | fveq2 6103 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → (𝐹‘𝑛) = (𝐹‘𝑘)) |
41 | 39, 40 | eleq12d 2682 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → ((𝑔‘𝑛) ∈ (𝐹‘𝑛) ↔ (𝑔‘𝑘) ∈ (𝐹‘𝑘))) |
42 | 41 | rspccva 3281 |
. . . . . . . . . . . . . 14
⊢
((∀𝑛 ∈
𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ 𝑘 ∈ 𝐼) → (𝑔‘𝑘) ∈ (𝐹‘𝑘)) |
43 | 38, 42 | sylan 487 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) ∧ 𝑘 ∈ 𝐼) → (𝑔‘𝑘) ∈ (𝐹‘𝑘)) |
44 | | simprrl 800 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → (𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛))) |
45 | 44 | simpld 474 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → 𝑤 ∈ Fin) |
46 | 44 | simprd 478 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) |
47 | 40 | unieqd 4382 |
. . . . . . . . . . . . . . . 16
⊢ (𝑛 = 𝑘 → ∪ (𝐹‘𝑛) = ∪ (𝐹‘𝑘)) |
48 | 39, 47 | eqeq12d 2625 |
. . . . . . . . . . . . . . 15
⊢ (𝑛 = 𝑘 → ((𝑔‘𝑛) = ∪ (𝐹‘𝑛) ↔ (𝑔‘𝑘) = ∪ (𝐹‘𝑘))) |
49 | 48 | rspccva 3281 |
. . . . . . . . . . . . . 14
⊢
((∀𝑛 ∈
(𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛) ∧ 𝑘 ∈ (𝐼 ∖ 𝑤)) → (𝑔‘𝑘) = ∪ (𝐹‘𝑘)) |
50 | 46, 49 | sylan 487 |
. . . . . . . . . . . . 13
⊢ (((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) ∧ 𝑘 ∈ (𝐼 ∖ 𝑤)) → (𝑔‘𝑘) = ∪ (𝐹‘𝑘)) |
51 | | simprrr 801 |
. . . . . . . . . . . . . 14
⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)) |
52 | 39 | cbvixpv 7812 |
. . . . . . . . . . . . . 14
⊢ X𝑛 ∈
𝐼 (𝑔‘𝑛) = X𝑘 ∈ 𝐼 (𝑔‘𝑘) |
53 | 51, 52 | syl6eleq 2698 |
. . . . . . . . . . . . 13
⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑘 ∈ 𝐼 (𝑔‘𝑘)) |
54 | 25, 1, 17, 3, 26, 8, 36, 37, 43, 45, 50, 53 | ptcnplem 21234 |
. . . . . . . . . . . 12
⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘))) |
55 | 54 | anassrs 678 |
. . . . . . . . . . 11
⊢ (((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛))) ∧ ((𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛))) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘))) |
56 | 55 | expr 641 |
. . . . . . . . . 10
⊢ (((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛))) ∧ (𝑤 ∈ Fin ∧ ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛))) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
57 | 56 | rexlimdvaa 3014 |
. . . . . . . . 9
⊢ ((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛))) → (∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘))))) |
58 | 57 | impr 647 |
. . . . . . . 8
⊢ ((𝜑 ∧ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛)) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛))) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
59 | 24, 58 | sylan2b 491 |
. . . . . . 7
⊢ ((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛))) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
60 | | eleq2 2677 |
. . . . . . . 8
⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 ↔ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛))) |
61 | 52 | eqeq2i 2622 |
. . . . . . . . . . . 12
⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) ↔ 𝑓 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)) |
62 | 61 | biimpi 205 |
. . . . . . . . . . 11
⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → 𝑓 = X𝑘 ∈ 𝐼 (𝑔‘𝑘)) |
63 | 62 | sseq2d 3596 |
. . . . . . . . . 10
⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓 ↔ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘))) |
64 | 63 | anbi2d 736 |
. . . . . . . . 9
⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ((𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓) ↔ (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
65 | 64 | rexbidv 3034 |
. . . . . . . 8
⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → (∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓) ↔ ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘)))) |
66 | 60, 65 | imbi12d 333 |
. . . . . . 7
⊢ (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ((((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓)) ↔ (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ X𝑛 ∈ 𝐼 (𝑔‘𝑛) → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ X𝑘 ∈ 𝐼 (𝑔‘𝑘))))) |
67 | 59, 66 | syl5ibrcom 236 |
. . . . . 6
⊢ ((𝜑 ∧ (𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛))) → (𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓)))) |
68 | 67 | expimpd 627 |
. . . . 5
⊢ (𝜑 → (((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓)))) |
69 | 68 | exlimdv 1848 |
. . . 4
⊢ (𝜑 → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓)))) |
70 | 69 | alrimiv 1842 |
. . 3
⊢ (𝜑 → ∀𝑓(∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓)))) |
71 | | eqeq1 2614 |
. . . . . 6
⊢ (𝑎 = 𝑓 → (𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛) ↔ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))) |
72 | 71 | anbi2d 736 |
. . . . 5
⊢ (𝑎 = 𝑓 → (((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)) ↔ ((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) |
73 | 72 | exbidv 1837 |
. . . 4
⊢ (𝑎 = 𝑓 → (∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)) ↔ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)))) |
74 | 73 | ralab 3334 |
. . 3
⊢
(∀𝑓 ∈
{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))} (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓)) ↔ ∀𝑓(∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑓 = X𝑛 ∈ 𝐼 (𝑔‘𝑛)) → (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓)))) |
75 | 70, 74 | sylibr 223 |
. 2
⊢ (𝜑 → ∀𝑓 ∈ {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))} (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓))) |
76 | | ffn 5958 |
. . . . . 6
⊢ (𝐹:𝐼⟶Top → 𝐹 Fn 𝐼) |
77 | 3, 76 | syl 17 |
. . . . 5
⊢ (𝜑 → 𝐹 Fn 𝐼) |
78 | | eqid 2610 |
. . . . . 6
⊢ {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))} = {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))} |
79 | 78 | ptval 21183 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ 𝐹 Fn 𝐼) → (∏t‘𝐹) = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))})) |
80 | 17, 77, 79 | syl2anc 691 |
. . . 4
⊢ (𝜑 →
(∏t‘𝐹) = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))})) |
81 | 25, 80 | syl5eq 2656 |
. . 3
⊢ (𝜑 → 𝐾 = (topGen‘{𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))})) |
82 | 3 | feqmptd 6159 |
. . . . . 6
⊢ (𝜑 → 𝐹 = (𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘))) |
83 | 82 | fveq2d 6107 |
. . . . 5
⊢ (𝜑 →
(∏t‘𝐹) = (∏t‘(𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘)))) |
84 | 25, 83 | syl5eq 2656 |
. . . 4
⊢ (𝜑 → 𝐾 = (∏t‘(𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘)))) |
85 | 7 | ralrimiva 2949 |
. . . . 5
⊢ (𝜑 → ∀𝑘 ∈ 𝐼 (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘))) |
86 | | eqid 2610 |
. . . . . 6
⊢
(∏t‘(𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘))) = (∏t‘(𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘))) |
87 | 86 | pttopon 21209 |
. . . . 5
⊢ ((𝐼 ∈ 𝑉 ∧ ∀𝑘 ∈ 𝐼 (𝐹‘𝑘) ∈ (TopOn‘∪ (𝐹‘𝑘))) → (∏t‘(𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘))) ∈ (TopOn‘X𝑘 ∈
𝐼 ∪ (𝐹‘𝑘))) |
88 | 17, 85, 87 | syl2anc 691 |
. . . 4
⊢ (𝜑 →
(∏t‘(𝑘 ∈ 𝐼 ↦ (𝐹‘𝑘))) ∈ (TopOn‘X𝑘 ∈
𝐼 ∪ (𝐹‘𝑘))) |
89 | 84, 88 | eqeltrd 2688 |
. . 3
⊢ (𝜑 → 𝐾 ∈ (TopOn‘X𝑘 ∈
𝐼 ∪ (𝐹‘𝑘))) |
90 | 1, 81, 89, 26 | tgcnp 20867 |
. 2
⊢ (𝜑 → ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷) ↔ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)):𝑋⟶X𝑘 ∈ 𝐼 ∪ (𝐹‘𝑘) ∧ ∀𝑓 ∈ {𝑎 ∣ ∃𝑔((𝑔 Fn 𝐼 ∧ ∀𝑛 ∈ 𝐼 (𝑔‘𝑛) ∈ (𝐹‘𝑛) ∧ ∃𝑤 ∈ Fin ∀𝑛 ∈ (𝐼 ∖ 𝑤)(𝑔‘𝑛) = ∪ (𝐹‘𝑛)) ∧ 𝑎 = X𝑛 ∈ 𝐼 (𝑔‘𝑛))} (((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴))‘𝐷) ∈ 𝑓 → ∃𝑧 ∈ 𝐽 (𝐷 ∈ 𝑧 ∧ ((𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) “ 𝑧) ⊆ 𝑓))))) |
91 | 23, 75, 90 | mpbir2and 959 |
1
⊢ (𝜑 → (𝑥 ∈ 𝑋 ↦ (𝑘 ∈ 𝐼 ↦ 𝐴)) ∈ ((𝐽 CnP 𝐾)‘𝐷)) |