Step | Hyp | Ref
| Expression |
1 | | 0ov 7250 |
. . 3
⊢ (𝐴∅𝐵) = ∅ |
2 | | df-wwlksnon 27916 |
. . . . 5
⊢
WWalksNOn = (𝑛 ∈
ℕ0, 𝑔
∈ V ↦ (𝑎 ∈
(Vtx‘𝑔), 𝑏 ∈ (Vtx‘𝑔) ↦ {𝑤 ∈ (𝑛 WWalksN 𝑔) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑛) = 𝑏)})) |
3 | 2 | mpondm0 7446 |
. . . 4
⊢ (¬
(𝑁 ∈
ℕ0 ∧ 𝐺
∈ V) → (𝑁
WWalksNOn 𝐺) =
∅) |
4 | 3 | oveqd 7230 |
. . 3
⊢ (¬
(𝑁 ∈
ℕ0 ∧ 𝐺
∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = (𝐴∅𝐵)) |
5 | | df-wwlksn 27915 |
. . . . . 6
⊢ WWalksN
= (𝑛 ∈
ℕ0, 𝑔
∈ V ↦ {𝑤 ∈
(WWalks‘𝑔) ∣
(♯‘𝑤) = (𝑛 + 1)}) |
6 | 5 | mpondm0 7446 |
. . . . 5
⊢ (¬
(𝑁 ∈
ℕ0 ∧ 𝐺
∈ V) → (𝑁 WWalksN
𝐺) =
∅) |
7 | 6 | rabeqdv 3395 |
. . . 4
⊢ (¬
(𝑁 ∈
ℕ0 ∧ 𝐺
∈ V) → {𝑤 ∈
(𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)}) |
8 | | rab0 4297 |
. . . 4
⊢ {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅ |
9 | 7, 8 | eqtrdi 2794 |
. . 3
⊢ (¬
(𝑁 ∈
ℕ0 ∧ 𝐺
∈ V) → {𝑤 ∈
(𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅) |
10 | 1, 4, 9 | 3eqtr4a 2804 |
. 2
⊢ (¬
(𝑁 ∈
ℕ0 ∧ 𝐺
∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)}) |
11 | | iswwlksnon.v |
. . . . . . . . 9
⊢ 𝑉 = (Vtx‘𝐺) |
12 | 11 | wwlksnon 27935 |
. . . . . . . 8
⊢ ((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) →
(𝑁 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})) |
13 | 12 | adantr 484 |
. . . . . . 7
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) ∧
¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑁 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})) |
14 | 13 | oveqd 7230 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) ∧
¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = (𝐴(𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})𝐵)) |
15 | | eqid 2737 |
. . . . . . . 8
⊢ (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)}) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)}) |
16 | 15 | mpondm0 7446 |
. . . . . . 7
⊢ (¬
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})𝐵) = ∅) |
17 | 16 | adantl 485 |
. . . . . 6
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) ∧
¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})𝐵) = ∅) |
18 | 14, 17 | eqtrd 2777 |
. . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) ∧
¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅) |
19 | 18 | ex 416 |
. . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) →
(¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)) |
20 | 4, 1 | eqtrdi 2794 |
. . . . 5
⊢ (¬
(𝑁 ∈
ℕ0 ∧ 𝐺
∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅) |
21 | 20 | a1d 25 |
. . . 4
⊢ (¬
(𝑁 ∈
ℕ0 ∧ 𝐺
∈ V) → (¬ (𝐴
∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)) |
22 | 19, 21 | pm2.61i 185 |
. . 3
⊢ (¬
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅) |
23 | 11 | wwlknllvtx 27930 |
. . . . . . 7
⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → ((𝑤‘0) ∈ 𝑉 ∧ (𝑤‘𝑁) ∈ 𝑉)) |
24 | | eleq1 2825 |
. . . . . . . . 9
⊢ (𝐴 = (𝑤‘0) → (𝐴 ∈ 𝑉 ↔ (𝑤‘0) ∈ 𝑉)) |
25 | 24 | eqcoms 2745 |
. . . . . . . 8
⊢ ((𝑤‘0) = 𝐴 → (𝐴 ∈ 𝑉 ↔ (𝑤‘0) ∈ 𝑉)) |
26 | | eleq1 2825 |
. . . . . . . . 9
⊢ (𝐵 = (𝑤‘𝑁) → (𝐵 ∈ 𝑉 ↔ (𝑤‘𝑁) ∈ 𝑉)) |
27 | 26 | eqcoms 2745 |
. . . . . . . 8
⊢ ((𝑤‘𝑁) = 𝐵 → (𝐵 ∈ 𝑉 ↔ (𝑤‘𝑁) ∈ 𝑉)) |
28 | 25, 27 | bi2anan9 639 |
. . . . . . 7
⊢ (((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ↔ ((𝑤‘0) ∈ 𝑉 ∧ (𝑤‘𝑁) ∈ 𝑉))) |
29 | 23, 28 | syl5ibrcom 250 |
. . . . . 6
⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) |
30 | 29 | con3rr3 158 |
. . . . 5
⊢ (¬
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑤 ∈ (𝑁 WWalksN 𝐺) → ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵))) |
31 | 30 | ralrimiv 3104 |
. . . 4
⊢ (¬
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)) |
32 | | rabeq0 4299 |
. . . 4
⊢ ({𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅ ↔ ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)) |
33 | 31, 32 | sylibr 237 |
. . 3
⊢ (¬
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅) |
34 | 22, 33 | eqtr4d 2780 |
. 2
⊢ (¬
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)}) |
35 | 12 | adantr 484 |
. . 3
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) ∧
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝑁 WWalksNOn 𝐺) = (𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})) |
36 | | eqeq2 2749 |
. . . . . 6
⊢ (𝑎 = 𝐴 → ((𝑤‘0) = 𝑎 ↔ (𝑤‘0) = 𝐴)) |
37 | | eqeq2 2749 |
. . . . . 6
⊢ (𝑏 = 𝐵 → ((𝑤‘𝑁) = 𝑏 ↔ (𝑤‘𝑁) = 𝐵)) |
38 | 36, 37 | bi2anan9 639 |
. . . . 5
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵))) |
39 | 38 | rabbidv 3390 |
. . . 4
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)}) |
40 | 39 | adantl 485 |
. . 3
⊢ ((((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) ∧
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) ∧ (𝑎 = 𝐴 ∧ 𝑏 = 𝐵)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)} = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)}) |
41 | | simprl 771 |
. . 3
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) ∧
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 ∈ 𝑉) |
42 | | simprr 773 |
. . 3
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) ∧
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐵 ∈ 𝑉) |
43 | | ovex 7246 |
. . . . 5
⊢ (𝑁 WWalksN 𝐺) ∈ V |
44 | 43 | rabex 5225 |
. . . 4
⊢ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} ∈ V |
45 | 44 | a1i 11 |
. . 3
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) ∧
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} ∈ V) |
46 | 35, 40, 41, 42, 45 | ovmpod 7361 |
. 2
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) ∧
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)}) |
47 | 10, 34, 46 | ecase 1033 |
1
⊢ (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} |