Definition df-trail 26037

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.)

Assertion

Ref	Expression
df-trail	⊢ Trails = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ Fun ◡𝑓)})

Distinct variable group:   𝑣,𝑒,𝑓,𝑝

Detailed syntax breakdown of Definition df-trail

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)})

This definition is referenced by:  trls  26066  trliswlk  26069