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Definition df-trail 25831
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
StepHypRef Expression
1 ctrail 25821 . 2 class Trails
2 vv . . 3 setvar 𝑣
3 ve . . 3 setvar 𝑒
4 cvv 3172 . . 3 class V
5 vf . . . . . . 7 setvar 𝑓
65cv 1473 . . . . . 6 class 𝑓
7 vp . . . . . . 7 setvar 𝑝
87cv 1473 . . . . . 6 class 𝑝
92cv 1473 . . . . . . 7 class 𝑣
103cv 1473 . . . . . . 7 class 𝑒
11 cwalk 25820 . . . . . . 7 class Walks
129, 10, 11co 6527 . . . . . 6 class (𝑣 Walks 𝑒)
136, 8, 12wbr 4577 . . . . 5 wff 𝑓(𝑣 Walks 𝑒)𝑝
146ccnv 5027 . . . . . 6 class 𝑓
1514wfun 5784 . . . . 5 wff Fun 𝑓
1613, 15wa 382 . . . 4 wff (𝑓(𝑣 Walks 𝑒)𝑝 ∧ Fun 𝑓)
1716, 5, 7copab 4636 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ Fun 𝑓)}
182, 3, 4, 4, 17cmpt2 6529 . 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ Fun 𝑓)})
191, 18wceq 1474 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  25860  trliswlk  25863
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