Definition df-trls 27962

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

Step	Hyp	Ref	Expression
1		ctrls 27960	. 2 class Trails
2		vg	. . 3 setvar 𝑔
3		cvv 3422	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1538	. . . . . 6 class 𝑓
6		vp	. . . . . . 7 setvar 𝑝
7	6	cv 1538	. . . . . 6 class 𝑝
8	2	cv 1538	. . . . . . 7 class 𝑔
9		cwlks 27866	. . . . . . 7 class Walks
10	8, 9	cfv 6418	. . . . . 6 class (Walks‘𝑔)
11	5, 7, 10	wbr 5070	. . . . 5 wff 𝑓(Walks‘𝑔)𝑝
12	5	ccnv 5579	. . . . . 6 class ◡𝑓
13	12	wfun 6412	. . . . 5 wff Fun ◡𝑓
14	11, 13	wa 395	. . . 4 wff (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)
15	14, 4, 6	copab 5132	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)}
16	2, 3, 15	cmpt 5153	. 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)})
17	1, 16	wceq 1539	1 wff Trails = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ Fun ◡𝑓)})

This definition is referenced by:  reltrls  27964  trlsfval  27965