| Step | Hyp | Ref
| Expression |
| 1 | | df-wspthsnon 29854 |
. . 3
⊢
WSPathsNOn = (𝑛 ∈
ℕ0, 𝑔
∈ V ↦ (𝑎 ∈
(Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤})) |
| 2 | 1 | a1i 11 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐺 ∈ 𝑈) → WSPathsNOn = (𝑛 ∈ ℕ0,
𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}))) |
| 3 | | fveq2 6906 |
. . . . . 6
⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) |
| 4 | | wwlksnon.v |
. . . . . 6
⊢ 𝑉 = (Vtx‘𝐺) |
| 5 | 3, 4 | eqtr4di 2795 |
. . . . 5
⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉) |
| 6 | 5 | adantl 481 |
. . . 4
⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (Vtx‘𝑔) = 𝑉) |
| 7 | | oveq12 7440 |
. . . . . 6
⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑛 WWalksNOn 𝑔) = (𝑁 WWalksNOn 𝐺)) |
| 8 | 7 | oveqd 7448 |
. . . . 5
⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑎(𝑛 WWalksNOn 𝑔)𝑏) = (𝑎(𝑁 WWalksNOn 𝐺)𝑏)) |
| 9 | | fveq2 6906 |
. . . . . . . . 9
⊢ (𝑔 = 𝐺 → (SPathsOn‘𝑔) = (SPathsOn‘𝐺)) |
| 10 | 9 | oveqd 7448 |
. . . . . . . 8
⊢ (𝑔 = 𝐺 → (𝑎(SPathsOn‘𝑔)𝑏) = (𝑎(SPathsOn‘𝐺)𝑏)) |
| 11 | 10 | breqd 5154 |
. . . . . . 7
⊢ (𝑔 = 𝐺 → (𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤 ↔ 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤)) |
| 12 | 11 | adantl 481 |
. . . . . 6
⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤 ↔ 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤)) |
| 13 | 12 | exbidv 1921 |
. . . . 5
⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤 ↔ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤)) |
| 14 | 8, 13 | rabeqbidv 3455 |
. . . 4
⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤} = {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}) |
| 15 | 6, 6, 14 | mpoeq123dv 7508 |
. . 3
⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})) |
| 16 | 15 | adantl 481 |
. 2
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ 𝑈) ∧ (𝑛 = 𝑁 ∧ 𝑔 = 𝐺)) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑎(𝑛 WWalksNOn 𝑔)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝑔)𝑏)𝑤}) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})) |
| 17 | | simpl 482 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐺 ∈ 𝑈) → 𝑁 ∈
ℕ0) |
| 18 | | elex 3501 |
. . 3
⊢ (𝐺 ∈ 𝑈 → 𝐺 ∈ V) |
| 19 | 18 | adantl 481 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐺 ∈ 𝑈) → 𝐺 ∈ V) |
| 20 | 4 | fvexi 6920 |
. . . 4
⊢ 𝑉 ∈ V |
| 21 | 20, 20 | mpoex 8104 |
. . 3
⊢ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}) ∈ V |
| 22 | 21 | a1i 11 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐺 ∈ 𝑈) → (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤}) ∈ V) |
| 23 | 2, 16, 17, 19, 22 | ovmpod 7585 |
1
⊢ ((𝑁 ∈ ℕ0
∧ 𝐺 ∈ 𝑈) → (𝑁 WSPathsNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑎(𝑁 WWalksNOn 𝐺)𝑏) ∣ ∃𝑓 𝑓(𝑎(SPathsOn‘𝐺)𝑏)𝑤})) |