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Mirrors > Home > MPE Home > Th. List > df-trls | 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.) (Revised by AV, 28-Dec-2020.) |
Ref | Expression |
---|---|
df-trls | ⊢ Trails = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ctrls 27802 | . 2 class Trails | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3421 | . . 3 class V | |
4 | vf | . . . . . . 7 setvar 𝑓 | |
5 | 4 | cv 1542 | . . . . . 6 class 𝑓 |
6 | vp | . . . . . . 7 setvar 𝑝 | |
7 | 6 | cv 1542 | . . . . . 6 class 𝑝 |
8 | 2 | cv 1542 | . . . . . . 7 class 𝑔 |
9 | cwlks 27708 | . . . . . . 7 class Walks | |
10 | 8, 9 | cfv 6398 | . . . . . 6 class (Walks‘𝑔) |
11 | 5, 7, 10 | wbr 5068 | . . . . 5 wff 𝑓(Walks‘𝑔)𝑝 |
12 | 5 | ccnv 5565 | . . . . . 6 class ◡𝑓 |
13 | 12 | wfun 6392 | . . . . 5 wff Fun ◡𝑓 |
14 | 11, 13 | wa 399 | . . . 4 wff (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓) |
15 | 14, 4, 6 | copab 5130 | . . 3 class {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)} |
16 | 2, 3, 15 | cmpt 5150 | . 2 class (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)}) |
17 | 1, 16 | wceq 1543 | 1 wff Trails = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)}) |
Colors of variables: wff setvar class |
This definition is referenced by: reltrls 27806 trlsfval 27807 |
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