Step | Hyp | Ref
| Expression |
1 | | simpll 786 |
. . . . . . . . 9
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝐵 ∈ 𝑉) → 𝑉 ∈ 𝑋) |
2 | | simpr 476 |
. . . . . . . . . 10
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → 𝐸 ∈ 𝑌) |
3 | 2 | adantr 480 |
. . . . . . . . 9
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝐵 ∈ 𝑉) → 𝐸 ∈ 𝑌) |
4 | | simpr 476 |
. . . . . . . . 9
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝐵 ∈ 𝑉) → 𝐵 ∈ 𝑉) |
5 | 1, 3, 4 | 3jca 1235 |
. . . . . . . 8
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝐵 ∈ 𝑉) → (𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐵 ∈ 𝑉)) |
6 | 5 | adantr 480 |
. . . . . . 7
⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝐵 ∈ 𝑉) ∧ (𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → (𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐵 ∈ 𝑉)) |
7 | | simpr1 1060 |
. . . . . . 7
⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝐵 ∈ 𝑉) ∧ (𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → 𝐼 ≠ 𝐽) |
8 | | 3simpc 1053 |
. . . . . . . 8
⊢ ((𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶}) → ((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) |
9 | 8 | adantl 481 |
. . . . . . 7
⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝐵 ∈ 𝑉) ∧ (𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → ((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) |
10 | | 2trlY.i |
. . . . . . . 8
⊢ (𝐼 ∈ 𝑈 ∧ 𝐽 ∈ 𝑊) |
11 | | 2trlY.f |
. . . . . . . 8
⊢ 𝐹 = {〈0, 𝐼〉, 〈1, 𝐽〉} |
12 | 10, 11 | 2trllemE 26083 |
. . . . . . 7
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌 ∧ 𝐵 ∈ 𝑉) ∧ 𝐼 ≠ 𝐽 ∧ ((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸) |
13 | 6, 7, 9, 12 | syl3anc 1318 |
. . . . . 6
⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝐵 ∈ 𝑉) ∧ (𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸) |
14 | 13 | ex 449 |
. . . . 5
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ 𝐵 ∈ 𝑉) → ((𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶}) → 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸)) |
15 | 14 | 3ad2antr2 1220 |
. . . 4
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶}) → 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸)) |
16 | 15 | imp 444 |
. . 3
⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → 𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸) |
17 | | 2trlY.p |
. . . . . . 7
⊢ 𝑃 = {〈0, 𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉} |
18 | 17 | 2trllemG 26088 |
. . . . . 6
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝑃:(0...2)⟶𝑉) |
19 | 10, 11 | 2trllemA 26080 |
. . . . . . . 8
⊢
(#‘𝐹) =
2 |
20 | 19 | oveq2i 6560 |
. . . . . . 7
⊢
(0...(#‘𝐹)) =
(0...2) |
21 | 20 | feq2i 5950 |
. . . . . 6
⊢ (𝑃:(0...(#‘𝐹))⟶𝑉 ↔ 𝑃:(0...2)⟶𝑉) |
22 | 18, 21 | sylibr 223 |
. . . . 5
⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉) → 𝑃:(0...(#‘𝐹))⟶𝑉) |
23 | 22 | adantl 481 |
. . . 4
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → 𝑃:(0...(#‘𝐹))⟶𝑉) |
24 | 23 | adantr 480 |
. . 3
⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → 𝑃:(0...(#‘𝐹))⟶𝑉) |
25 | 10, 11, 17 | 2wlklem1 26127 |
. . . . 5
⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ ((𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → ∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) |
26 | 8, 25 | sylan2 490 |
. . . 4
⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → ∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) |
27 | 10, 11 | 2trllemB 26081 |
. . . . . 6
⊢
(0..^(#‘𝐹)) =
{0, 1} |
28 | 27 | a1i 11 |
. . . . 5
⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → (0..^(#‘𝐹)) = {0, 1}) |
29 | 28 | raleqdv 3121 |
. . . 4
⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → (∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))} ↔ ∀𝑘 ∈ {0, 1} (𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))})) |
30 | 26, 29 | mpbird 246 |
. . 3
⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}) |
31 | | prex 4836 |
. . . . . . 7
⊢ {〈0,
𝐼〉, 〈1, 𝐽〉} ∈
V |
32 | 11, 31 | eqeltri 2684 |
. . . . . 6
⊢ 𝐹 ∈ V |
33 | | tpex 6855 |
. . . . . . 7
⊢ {〈0,
𝐴〉, 〈1, 𝐵〉, 〈2, 𝐶〉} ∈
V |
34 | 17, 33 | eqeltri 2684 |
. . . . . 6
⊢ 𝑃 ∈ V |
35 | | istrl2 26068 |
. . . . . 6
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐹 ∈ V ∧ 𝑃 ∈ V)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
36 | 32, 34, 35 | mpanr12 717 |
. . . . 5
⊢ ((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
37 | 36 | adantr 480 |
. . . 4
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
38 | 37 | adantr 480 |
. . 3
⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → (𝐹(𝑉 Trails 𝐸)𝑃 ↔ (𝐹:(0..^(#‘𝐹))–1-1→dom 𝐸 ∧ 𝑃:(0...(#‘𝐹))⟶𝑉 ∧ ∀𝑘 ∈ (0..^(#‘𝐹))(𝐸‘(𝐹‘𝑘)) = {(𝑃‘𝑘), (𝑃‘(𝑘 + 1))}))) |
39 | 16, 24, 30, 38 | mpbir3and 1238 |
. 2
⊢ ((((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) ∧ (𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶})) → 𝐹(𝑉 Trails 𝐸)𝑃) |
40 | 39 | ex 449 |
1
⊢ (((𝑉 ∈ 𝑋 ∧ 𝐸 ∈ 𝑌) ∧ (𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑉 ∧ 𝐶 ∈ 𝑉)) → ((𝐼 ≠ 𝐽 ∧ (𝐸‘𝐼) = {𝐴, 𝐵} ∧ (𝐸‘𝐽) = {𝐵, 𝐶}) → 𝐹(𝑉 Trails 𝐸)𝑃)) |