Step | Hyp | Ref
| Expression |
1 | | simpr 476 |
. . . . . . . . . . . . 13
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅) |
2 | 1 | unieqd 4382 |
. . . . . . . . . . . 12
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ∪ 𝑟 = ∪
𝑅) |
3 | | xkoval.x |
. . . . . . . . . . . 12
⊢ 𝑋 = ∪
𝑅 |
4 | 2, 3 | syl6eqr 2662 |
. . . . . . . . . . 11
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ∪ 𝑟 = 𝑋) |
5 | 4 | pweqd 4113 |
. . . . . . . . . 10
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝒫 ∪ 𝑟 =
𝒫 𝑋) |
6 | 1 | oveq1d 6564 |
. . . . . . . . . . 11
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑟 ↾t 𝑥) = (𝑅 ↾t 𝑥)) |
7 | 6 | eleq1d 2672 |
. . . . . . . . . 10
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((𝑟 ↾t 𝑥) ∈ Comp ↔ (𝑅 ↾t 𝑥) ∈ Comp)) |
8 | 5, 7 | rabeqbidv 3168 |
. . . . . . . . 9
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑥 ∈ 𝒫 ∪ 𝑟
∣ (𝑟
↾t 𝑥)
∈ Comp} = {𝑥 ∈
𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp}) |
9 | | xkoval.k |
. . . . . . . . 9
⊢ 𝐾 = {𝑥 ∈ 𝒫 𝑋 ∣ (𝑅 ↾t 𝑥) ∈ Comp} |
10 | 8, 9 | syl6eqr 2662 |
. . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑥 ∈ 𝒫 ∪ 𝑟
∣ (𝑟
↾t 𝑥)
∈ Comp} = 𝐾) |
11 | | simpl 472 |
. . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑠 = 𝑆) |
12 | 1, 11 | oveq12d 6567 |
. . . . . . . . 9
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑟 Cn 𝑠) = (𝑅 Cn 𝑆)) |
13 | | rabeq 3166 |
. . . . . . . . 9
⊢ ((𝑟 Cn 𝑠) = (𝑅 Cn 𝑆) → {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) |
14 | 12, 13 | syl 17 |
. . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) |
15 | 10, 11, 14 | mpt2eq123dv 6615 |
. . . . . . 7
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟
∣ (𝑟
↾t 𝑥)
∈ Comp}, 𝑣 ∈
𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) |
16 | | xkoval.t |
. . . . . . 7
⊢ 𝑇 = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) |
17 | 15, 16 | syl6eqr 2662 |
. . . . . 6
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟
∣ (𝑟
↾t 𝑥)
∈ Comp}, 𝑣 ∈
𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = 𝑇) |
18 | 17 | rneqd 5274 |
. . . . 5
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟
∣ (𝑟
↾t 𝑥)
∈ Comp}, 𝑣 ∈
𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = ran 𝑇) |
19 | 18 | fveq2d 6107 |
. . . 4
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟
∣ (𝑟
↾t 𝑥)
∈ Comp}, 𝑣 ∈
𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})) = (fi‘ran 𝑇)) |
20 | 19 | fveq2d 6107 |
. . 3
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟
∣ (𝑟
↾t 𝑥)
∈ Comp}, 𝑣 ∈
𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}))) = (topGen‘(fi‘ran 𝑇))) |
21 | | df-xko 21176 |
. . 3
⊢
^ko = (𝑠
∈ Top, 𝑟 ∈ Top
↦ (topGen‘(fi‘ran (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟
∣ (𝑟
↾t 𝑥)
∈ Comp}, 𝑣 ∈
𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣})))) |
22 | | fvex 6113 |
. . 3
⊢
(topGen‘(fi‘ran 𝑇)) ∈ V |
23 | 20, 21, 22 | ovmpt2a 6689 |
. 2
⊢ ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran
𝑇))) |
24 | 23 | ancoms 468 |
1
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ^ko 𝑅) = (topGen‘(fi‘ran
𝑇))) |