| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | simpr 484 | . . . . . . . . . . . . 13
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑟 = 𝑅) | 
| 2 | 1 | unieqd 4920 | . . . . . . . . . . . 12
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ∪ 𝑟 = ∪
𝑅) | 
| 3 |  | xkoval.x | . . . . . . . . . . . 12
⊢ 𝑋 = ∪
𝑅 | 
| 4 | 2, 3 | eqtr4di 2795 | . . . . . . . . . . 11
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ∪ 𝑟 = 𝑋) | 
| 5 | 4 | pweqd 4617 | . . . . . . . . . 10
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝒫 ∪ 𝑟 =
𝒫 𝑋) | 
| 6 | 1 | oveq1d 7446 | . . . . . . . . . . 11
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑟 ↾t 𝑥) = (𝑅 ↾t 𝑥)) | 
| 7 | 6 | eleq1d 2826 | . . . . . . . . . 10
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → ((𝑟 ↾t 𝑥) ∈ Comp ↔ (𝑅 ↾t 𝑥) ∈ Comp)) | 
| 8 | 5, 7 | rabeqbidv 3455 | . . . . . . . . 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 | eqtr4di 2795 | . . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑥 ∈ 𝒫 ∪ 𝑟
∣ (𝑟
↾t 𝑥)
∈ Comp} = 𝐾) | 
| 11 |  | simpl 482 | . . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → 𝑠 = 𝑆) | 
| 12 | 1, 11 | oveq12d 7449 | . . . . . . . . 9
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑟 Cn 𝑠) = (𝑅 Cn 𝑆)) | 
| 13 | 12 | rabeqdv 3452 | . . . . . . . 8
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣} = {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) | 
| 14 | 10, 11, 13 | mpoeq123dv 7508 | . . . . . . 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})) | 
| 15 |  | xkoval.t | . . . . . . 7
⊢ 𝑇 = (𝑘 ∈ 𝐾, 𝑣 ∈ 𝑆 ↦ {𝑓 ∈ (𝑅 Cn 𝑆) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) | 
| 16 | 14, 15 | eqtr4di 2795 | . . . . . 6
⊢ ((𝑠 = 𝑆 ∧ 𝑟 = 𝑅) → (𝑘 ∈ {𝑥 ∈ 𝒫 ∪ 𝑟
∣ (𝑟
↾t 𝑥)
∈ Comp}, 𝑣 ∈
𝑠 ↦ {𝑓 ∈ (𝑟 Cn 𝑠) ∣ (𝑓 “ 𝑘) ⊆ 𝑣}) = 𝑇) | 
| 17 | 16 | rneqd 5949 | . . . . 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) | 
| 18 | 17 | fveq2d 6910 | . . . 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)) | 
| 19 | 18 | fveq2d 6910 | . . 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))) | 
| 20 |  | df-xko 23571 | . . 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})))) | 
| 21 |  | fvex 6919 | . . 3
⊢
(topGen‘(fi‘ran 𝑇)) ∈ V | 
| 22 | 19, 20, 21 | ovmpoa 7588 | . 2
⊢ ((𝑆 ∈ Top ∧ 𝑅 ∈ Top) → (𝑆 ↑ko 𝑅) = (topGen‘(fi‘ran
𝑇))) | 
| 23 | 22 | ancoms 458 | 1
⊢ ((𝑅 ∈ Top ∧ 𝑆 ∈ Top) → (𝑆 ↑ko 𝑅) = (topGen‘(fi‘ran
𝑇))) |