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Mirrors > Home > MPE Home > Th. List > df-trail | Structured version Visualization version GIF version |
Description: Define the set of all
Trails (in an undirected graph).
According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A trail is a walk in which all edges are distinct. According to Bollobas: "... walk is called a trail if all its edges are distinct.", see Definition of [Bollobas] p. 5. Therefore, a trail can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the trail is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) |
Ref | Expression |
---|---|
df-trail | ⊢ Trails = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ Fun ◡𝑓)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ctrail 26027 | . 2 class Trails | |
2 | vv | . . 3 setvar 𝑣 | |
3 | ve | . . 3 setvar 𝑒 | |
4 | cvv 3173 | . . 3 class V | |
5 | vf | . . . . . . 7 setvar 𝑓 | |
6 | 5 | cv 1474 | . . . . . 6 class 𝑓 |
7 | vp | . . . . . . 7 setvar 𝑝 | |
8 | 7 | cv 1474 | . . . . . 6 class 𝑝 |
9 | 2 | cv 1474 | . . . . . . 7 class 𝑣 |
10 | 3 | cv 1474 | . . . . . . 7 class 𝑒 |
11 | cwalk 26026 | . . . . . . 7 class Walks | |
12 | 9, 10, 11 | co 6549 | . . . . . 6 class (𝑣 Walks 𝑒) |
13 | 6, 8, 12 | wbr 4583 | . . . . 5 wff 𝑓(𝑣 Walks 𝑒)𝑝 |
14 | 6 | ccnv 5037 | . . . . . 6 class ◡𝑓 |
15 | 14 | wfun 5798 | . . . . 5 wff Fun ◡𝑓 |
16 | 13, 15 | wa 383 | . . . 4 wff (𝑓(𝑣 Walks 𝑒)𝑝 ∧ Fun ◡𝑓) |
17 | 16, 5, 7 | copab 4642 | . . 3 class {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ Fun ◡𝑓)} |
18 | 2, 3, 4, 4, 17 | cmpt2 6551 | . 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ Fun ◡𝑓)}) |
19 | 1, 18 | wceq 1475 | 1 wff Trails = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ Fun ◡𝑓)}) |
Colors of variables: wff setvar class |
This definition is referenced by: trls 26066 trliswlk 26069 |
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