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Definition df-spths 40923
Description: Define the set of all simple paths (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

Therefore, a simple path can be represented by an injective mapping f from { 1 , ... , n } and an injective mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the simple path 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, 20-Oct-2017.) (Revised by AV, 9-Jan-2021.)

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

Detailed syntax breakdown of Definition df-spths
StepHypRef Expression
1 cspths 40919 . 2 class SPathS
2 vg . . 3 setvar 𝑔
3 cvv 3169 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1473 . . . . . 6 class 𝑓
6 vp . . . . . . 7 setvar 𝑝
76cv 1473 . . . . . 6 class 𝑝
82cv 1473 . . . . . . 7 class 𝑔
9 ctrls 40898 . . . . . . 7 class TrailS
108, 9cfv 5787 . . . . . 6 class (TrailS‘𝑔)
115, 7, 10wbr 4574 . . . . 5 wff 𝑓(TrailS‘𝑔)𝑝
127ccnv 5024 . . . . . 6 class 𝑝
1312wfun 5781 . . . . 5 wff Fun 𝑝
1411, 13wa 382 . . . 4 wff (𝑓(TrailS‘𝑔)𝑝 ∧ Fun 𝑝)
1514, 4, 6copab 4633 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(TrailS‘𝑔)𝑝 ∧ Fun 𝑝)}
162, 3, 15cmpt 4634 . 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(TrailS‘𝑔)𝑝 ∧ Fun 𝑝)})
171, 16wceq 1474 1 wff SPathS = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(TrailS‘𝑔)𝑝 ∧ Fun 𝑝)})
Colors of variables: wff setvar class
This definition is referenced by:  spthsfval  40927
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