| Step | Hyp | Ref | Expression | 
|---|
| 1 |  | 0ov 7469 | . . 3
⊢ (𝐴∅𝐵) = ∅ | 
| 2 |  | df-wwlksnon 29853 | . . . . 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 7674 | . . . 4
⊢ (¬
(𝑁 ∈
ℕ0 ∧ 𝐺
∈ V) → (𝑁
WWalksNOn 𝐺) =
∅) | 
| 4 | 3 | oveqd 7449 | . . 3
⊢ (¬
(𝑁 ∈
ℕ0 ∧ 𝐺
∈ V) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = (𝐴∅𝐵)) | 
| 5 |  | df-wwlksn 29852 | . . . . . 6
⊢  WWalksN
= (𝑛 ∈
ℕ0, 𝑔
∈ V ↦ {𝑤 ∈
(WWalks‘𝑔) ∣
(♯‘𝑤) = (𝑛 + 1)}) | 
| 6 | 5 | mpondm0 7674 | . . . . 5
⊢ (¬
(𝑁 ∈
ℕ0 ∧ 𝐺
∈ V) → (𝑁 WWalksN
𝐺) =
∅) | 
| 7 | 6 | rabeqdv 3451 | . . . 4
⊢ (¬
(𝑁 ∈
ℕ0 ∧ 𝐺
∈ V) → {𝑤 ∈
(𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)}) | 
| 8 |  | rab0 4385 | . . . 4
⊢ {𝑤 ∈ ∅ ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅ | 
| 9 | 7, 8 | eqtrdi 2792 | . . 3
⊢ (¬
(𝑁 ∈
ℕ0 ∧ 𝐺
∈ V) → {𝑤 ∈
(𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅) | 
| 10 | 1, 4, 9 | 3eqtr4a 2802 | . 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 29872 | . . . . . . . 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 480 | . . . . . . 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 7449 | . . . . . 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 2736 | . . . . . . . 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 7674 | . . . . . . 7
⊢ (¬
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑎 ∈ 𝑉, 𝑏 ∈ 𝑉 ↦ {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏)})𝐵) = ∅) | 
| 17 | 16 | adantl 481 | . . . . . 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 2776 | . . . . 5
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) ∧
¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅) | 
| 19 | 18 | ex 412 | . . . 4
⊢ ((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) →
(¬ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅)) | 
| 20 | 4, 1 | eqtrdi 2792 | . . . . 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 182 | . . 3
⊢ (¬
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = ∅) | 
| 23 | 11 | wwlknllvtx 29867 | . . . . . . 7
⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → ((𝑤‘0) ∈ 𝑉 ∧ (𝑤‘𝑁) ∈ 𝑉)) | 
| 24 |  | eleq1 2828 | . . . . . . . . 9
⊢ (𝐴 = (𝑤‘0) → (𝐴 ∈ 𝑉 ↔ (𝑤‘0) ∈ 𝑉)) | 
| 25 | 24 | eqcoms 2744 | . . . . . . . 8
⊢ ((𝑤‘0) = 𝐴 → (𝐴 ∈ 𝑉 ↔ (𝑤‘0) ∈ 𝑉)) | 
| 26 |  | eleq1 2828 | . . . . . . . . 9
⊢ (𝐵 = (𝑤‘𝑁) → (𝐵 ∈ 𝑉 ↔ (𝑤‘𝑁) ∈ 𝑉)) | 
| 27 | 26 | eqcoms 2744 | . . . . . . . 8
⊢ ((𝑤‘𝑁) = 𝐵 → (𝐵 ∈ 𝑉 ↔ (𝑤‘𝑁) ∈ 𝑉)) | 
| 28 | 25, 27 | bi2anan9 638 | . . . . . . 7
⊢ (((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵) → ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) ↔ ((𝑤‘0) ∈ 𝑉 ∧ (𝑤‘𝑁) ∈ 𝑉))) | 
| 29 | 23, 28 | syl5ibrcom 247 | . . . . . 6
⊢ (𝑤 ∈ (𝑁 WWalksN 𝐺) → (((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉))) | 
| 30 | 29 | con3rr3 155 | . . . . 5
⊢ (¬
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝑤 ∈ (𝑁 WWalksN 𝐺) → ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵))) | 
| 31 | 30 | ralrimiv 3144 | . . . 4
⊢ (¬
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → ∀𝑤 ∈ (𝑁 WWalksN 𝐺) ¬ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)) | 
| 32 |  | rabeq0 4387 | . . . 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 234 | . . 3
⊢ (¬
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)} = ∅) | 
| 34 | 22, 33 | eqtr4d 2779 | . 2
⊢ (¬
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉) → (𝐴(𝑁 WWalksNOn 𝐺)𝐵) = {𝑤 ∈ (𝑁 WWalksN 𝐺) ∣ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵)}) | 
| 35 | 12 | adantr 480 | . . 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 2748 | . . . . . 6
⊢ (𝑎 = 𝐴 → ((𝑤‘0) = 𝑎 ↔ (𝑤‘0) = 𝐴)) | 
| 37 |  | eqeq2 2748 | . . . . . 6
⊢ (𝑏 = 𝐵 → ((𝑤‘𝑁) = 𝑏 ↔ (𝑤‘𝑁) = 𝐵)) | 
| 38 | 36, 37 | bi2anan9 638 | . . . . 5
⊢ ((𝑎 = 𝐴 ∧ 𝑏 = 𝐵) → (((𝑤‘0) = 𝑎 ∧ (𝑤‘𝑁) = 𝑏) ↔ ((𝑤‘0) = 𝐴 ∧ (𝑤‘𝑁) = 𝐵))) | 
| 39 | 38 | rabbidv 3443 | . . . 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 481 | . . 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 770 | . . 3
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) ∧
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐴 ∈ 𝑉) | 
| 42 |  | simprr 772 | . . 3
⊢ (((𝑁 ∈ ℕ0
∧ 𝐺 ∈ V) ∧
(𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉)) → 𝐵 ∈ 𝑉) | 
| 43 |  | ovex 7465 | . . . . 5
⊢ (𝑁 WWalksN 𝐺) ∈ V | 
| 44 | 43 | rabex 5338 | . . . 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 7586 | . 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)} |