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Definition df-trls 26720
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.)

Assertion
Ref Expression
df-trls Trails = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun 𝑓)})
Distinct variable group:   𝑓,𝑔,𝑝

Detailed syntax breakdown of Definition df-trls
StepHypRef Expression
1 ctrls 26718 . 2 class Trails
2 vg . . 3 setvar 𝑔
3 cvv 3304 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1595 . . . . . 6 class 𝑓
6 vp . . . . . . 7 setvar 𝑝
76cv 1595 . . . . . 6 class 𝑝
82cv 1595 . . . . . . 7 class 𝑔
9 cwlks 26623 . . . . . . 7 class Walks
108, 9cfv 6001 . . . . . 6 class (Walks‘𝑔)
115, 7, 10wbr 4760 . . . . 5 wff 𝑓(Walks‘𝑔)𝑝
125ccnv 5217 . . . . . 6 class 𝑓
1312wfun 5995 . . . . 5 wff Fun 𝑓
1411, 13wa 383 . . . 4 wff (𝑓(Walks‘𝑔)𝑝 ∧ Fun 𝑓)
1514, 4, 6copab 4820 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun 𝑓)}
162, 3, 15cmpt 4837 . 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun 𝑓)})
171, 16wceq 1596 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  26722  trlsfval  26723
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