| Step | Hyp | Ref
| Expression |
| 1 | | df-wwlks 29850 |
. . 3
⊢ WWalks =
(𝑔 ∈ V ↦ {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))}) |
| 2 | | fveq2 6906 |
. . . . . 6
⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = (Vtx‘𝐺)) |
| 3 | | wwlks.v |
. . . . . 6
⊢ 𝑉 = (Vtx‘𝐺) |
| 4 | 2, 3 | eqtr4di 2795 |
. . . . 5
⊢ (𝑔 = 𝐺 → (Vtx‘𝑔) = 𝑉) |
| 5 | | wrdeq 14574 |
. . . . 5
⊢
((Vtx‘𝑔) =
𝑉 → Word
(Vtx‘𝑔) = Word 𝑉) |
| 6 | 4, 5 | syl 17 |
. . . 4
⊢ (𝑔 = 𝐺 → Word (Vtx‘𝑔) = Word 𝑉) |
| 7 | | fveq2 6906 |
. . . . . . . 8
⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = (Edg‘𝐺)) |
| 8 | | wwlks.e |
. . . . . . . 8
⊢ 𝐸 = (Edg‘𝐺) |
| 9 | 7, 8 | eqtr4di 2795 |
. . . . . . 7
⊢ (𝑔 = 𝐺 → (Edg‘𝑔) = 𝐸) |
| 10 | 9 | eleq2d 2827 |
. . . . . 6
⊢ (𝑔 = 𝐺 → ({(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ↔ {(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)) |
| 11 | 10 | ralbidv 3178 |
. . . . 5
⊢ (𝑔 = 𝐺 → (∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔) ↔ ∀𝑖 ∈ (0..^((♯‘𝑤) − 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)) |
| 12 | 11 | anbi2d 630 |
. . . 4
⊢ (𝑔 = 𝐺 → ((𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔)) ↔ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸))) |
| 13 | 6, 12 | rabeqbidv 3455 |
. . 3
⊢ (𝑔 = 𝐺 → {𝑤 ∈ Word (Vtx‘𝑔) ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ (Edg‘𝑔))} = {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)}) |
| 14 | | id 22 |
. . 3
⊢ (𝐺 ∈ V → 𝐺 ∈ V) |
| 15 | 3 | fvexi 6920 |
. . . . 5
⊢ 𝑉 ∈ V |
| 16 | 15 | a1i 11 |
. . . 4
⊢ (𝐺 ∈ V → 𝑉 ∈ V) |
| 17 | | wrdexg 14562 |
. . . 4
⊢ (𝑉 ∈ V → Word 𝑉 ∈ V) |
| 18 | | rabexg 5337 |
. . . 4
⊢ (Word
𝑉 ∈ V → {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)} ∈ V) |
| 19 | 16, 17, 18 | 3syl 18 |
. . 3
⊢ (𝐺 ∈ V → {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)} ∈ V) |
| 20 | 1, 13, 14, 19 | fvmptd3 7039 |
. 2
⊢ (𝐺 ∈ V →
(WWalks‘𝐺) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)}) |
| 21 | | fvprc 6898 |
. . 3
⊢ (¬
𝐺 ∈ V →
(WWalks‘𝐺) =
∅) |
| 22 | | fvprc 6898 |
. . . . . . . . . 10
⊢ (¬
𝐺 ∈ V →
(Vtx‘𝐺) =
∅) |
| 23 | 3, 22 | eqtrid 2789 |
. . . . . . . . 9
⊢ (¬
𝐺 ∈ V → 𝑉 = ∅) |
| 24 | | wrdeq 14574 |
. . . . . . . . 9
⊢ (𝑉 = ∅ → Word 𝑉 = Word
∅) |
| 25 | 23, 24 | syl 17 |
. . . . . . . 8
⊢ (¬
𝐺 ∈ V → Word
𝑉 = Word
∅) |
| 26 | 25 | eleq2d 2827 |
. . . . . . 7
⊢ (¬
𝐺 ∈ V → (𝑤 ∈ Word 𝑉 ↔ 𝑤 ∈ Word ∅)) |
| 27 | | 0wrd0 14578 |
. . . . . . 7
⊢ (𝑤 ∈ Word ∅ ↔
𝑤 =
∅) |
| 28 | 26, 27 | bitrdi 287 |
. . . . . 6
⊢ (¬
𝐺 ∈ V → (𝑤 ∈ Word 𝑉 ↔ 𝑤 = ∅)) |
| 29 | | nne 2944 |
. . . . . . . 8
⊢ (¬
𝑤 ≠ ∅ ↔ 𝑤 = ∅) |
| 30 | 29 | biimpri 228 |
. . . . . . 7
⊢ (𝑤 = ∅ → ¬ 𝑤 ≠ ∅) |
| 31 | 30 | intnanrd 489 |
. . . . . 6
⊢ (𝑤 = ∅ → ¬ (𝑤 ≠ ∅ ∧
∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)) |
| 32 | 28, 31 | biimtrdi 253 |
. . . . 5
⊢ (¬
𝐺 ∈ V → (𝑤 ∈ Word 𝑉 → ¬ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸))) |
| 33 | 32 | ralrimiv 3145 |
. . . 4
⊢ (¬
𝐺 ∈ V →
∀𝑤 ∈ Word 𝑉 ¬ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)) |
| 34 | | rabeq0 4388 |
. . . 4
⊢ ({𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)} = ∅ ↔ ∀𝑤 ∈ Word 𝑉 ¬ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)) |
| 35 | 33, 34 | sylibr 234 |
. . 3
⊢ (¬
𝐺 ∈ V → {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)} = ∅) |
| 36 | 21, 35 | eqtr4d 2780 |
. 2
⊢ (¬
𝐺 ∈ V →
(WWalks‘𝐺) = {𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)}) |
| 37 | 20, 36 | pm2.61i 182 |
1
⊢
(WWalks‘𝐺) =
{𝑤 ∈ Word 𝑉 ∣ (𝑤 ≠ ∅ ∧ ∀𝑖 ∈
(0..^((♯‘𝑤)
− 1)){(𝑤‘𝑖), (𝑤‘(𝑖 + 1))} ∈ 𝐸)} |