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Definition df-spth 26039
 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.)
Assertion
Ref Expression
df-spth SPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun 𝑝)})
Distinct variable group:   𝑣,𝑒,𝑓,𝑝

Detailed syntax breakdown of Definition df-spth
StepHypRef Expression
1 cspath 26029 . 2 class SPaths
2 vv . . 3 setvar 𝑣
3 ve . . 3 setvar 𝑒
4 cvv 3173 . . 3 class V
5 vf . . . . . . 7 setvar 𝑓
65cv 1474 . . . . . 6 class 𝑓
7 vp . . . . . . 7 setvar 𝑝
87cv 1474 . . . . . 6 class 𝑝
92cv 1474 . . . . . . 7 class 𝑣
103cv 1474 . . . . . . 7 class 𝑒
11 ctrail 26027 . . . . . . 7 class Trails
129, 10, 11co 6549 . . . . . 6 class (𝑣 Trails 𝑒)
136, 8, 12wbr 4583 . . . . 5 wff 𝑓(𝑣 Trails 𝑒)𝑝
148ccnv 5037 . . . . . 6 class 𝑝
1514wfun 5798 . . . . 5 wff Fun 𝑝
1613, 15wa 383 . . . 4 wff (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun 𝑝)
1716, 5, 7copab 4642 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun 𝑝)}
182, 3, 4, 4, 17cmpt2 6551 . 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun 𝑝)})
191, 18wceq 1475 1 wff SPaths = (𝑣 ∈ V, 𝑒 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(𝑣 Trails 𝑒)𝑝 ∧ Fun 𝑝)})
 Colors of variables: wff setvar class This definition is referenced by:  spths  26097  spthispth  26103
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