Step | Hyp | Ref
| Expression |
1 | | df-wwlksnon 26724 |
. . 3
⊢
WWalksNOn = (𝑛 ∈
ℕ0, 𝑔
∈ V ↦ (𝑎 ∈
(Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)})) |
2 | 1 | a1i 11 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐺 ∈ 𝑈) → WWalksNOn = (𝑛 ∈ ℕ0,
𝑔 ∈ V ↦ (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)}))) |
3 | | fveq2 6191 |
. . . . . 6
⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) |
4 | | wwlksnon.v |
. . . . . 6
⊢ 𝑉 = (Vtx‘𝐺) |
5 | 3, 4 | syl6eqr 2674 |
. . . . 5
⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉) |
6 | 5 | adantl 482 |
. . . 4
⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (Vtx‘𝑔) = 𝑉) |
7 | | oveq12 6659 |
. . . . 5
⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑛 WWalksN 𝑔) = (𝑁 WWalksN 𝐺)) |
8 | | fveq2 6191 |
. . . . . . . 8
⊢ (𝑛 = 𝑁 → (𝑤‘𝑛) = (𝑤‘𝑁)) |
9 | 8 | eqeq1d 2624 |
. . . . . . 7
⊢ (𝑛 = 𝑁 → ((𝑤‘𝑛) = 𝑏 ↔ (𝑤‘𝑁) = 𝑏)) |
10 | 9 | anbi2d 740 |
. . . . . 6
⊢ (𝑛 = 𝑁 → (((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏) ↔ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏))) |
11 | 10 | adantr 481 |
. . . . 5
⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏) ↔ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏))) |
12 | 7, 11 | rabeqbidv 3195 |
. . . 4
⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)}) |
13 | 6, 6, 12 | mpt2eq123dv 6717 |
. . 3
⊢ ((𝑛 = 𝑁 ∧ 𝑔 = 𝐺) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)}) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})) |
14 | 13 | adantl 482 |
. 2
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ 𝑈) ∧ (𝑛 = 𝑁 ∧ 𝑔 = 𝐺)) → (𝑎 ∈ (Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)}) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})) |
15 | | simpl 473 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐺 ∈ 𝑈) → 𝑁 ∈
ℕ0) |
16 | | elex 3212 |
. . 3
⊢ (𝐺 ∈ 𝑈 → 𝐺 ∈ V) |
17 | 16 | adantl 482 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐺 ∈ 𝑈) → 𝐺 ∈ V) |
18 | | fvex 6201 |
. . . . 5
⊢
(Vtx‘𝐺) ∈
V |
19 | 4, 18 | eqeltri 2697 |
. . . 4
⊢ 𝑉 ∈ V |
20 | 19, 19 | mpt2ex 7247 |
. . 3
⊢ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)}) ∈ V |
21 | 20 | a1i 11 |
. 2
⊢ ((𝑁 ∈ ℕ0
∧ 𝐺 ∈ 𝑈) → (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)}) ∈ V) |
22 | 2, 14, 15, 17, 21 | ovmpt2d 6788 |
1
⊢ ((𝑁 ∈ ℕ0
∧ 𝐺 ∈ 𝑈) → (𝑁 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})) |