Definition df-spths 28085

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

Step	Hyp	Ref	Expression
1		cspths 28081	. 2 class SPaths
2		vg	. . 3 setvar 𝑔
3		cvv 3432	. . 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		ctrls 28058	. . . . . . 7 class Trails
10	8, 9	cfv 6433	. . . . . 6 class (Trails‘𝑔)
11	5, 7, 10	wbr 5074	. . . . 5 wff 𝑓(Trails‘𝑔)𝑝
12	7	ccnv 5588	. . . . . 6 class ◡𝑝
13	12	wfun 6427	. . . . 5 wff Fun ◡𝑝
14	11, 13	wa 396	. . . 4 wff (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡𝑝)
15	14, 4, 6	copab 5136	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡𝑝)}
16	2, 3, 15	cmpt 5157	. 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡𝑝)})
17	1, 16	wceq 1539	1 wff SPaths = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Trails‘𝑔)𝑝 ∧ Fun ◡𝑝)})

This definition is referenced by:  spthsfval  28090