Definition df-clwlks 29025

Description: Define the set of all closed walks (in an undirected graph).

According to definition 4 in [Huneke] p. 2: "A walk of length n on (a graph) G is an ordered sequence v0 , v1 , ... v(n) of vertices such that v(i) and v(i+1) are neighbors (i.e are connected by an edge). We say the walk is closed if v(n) = v0".

According to the definition of a walk as two mappings f from { 0 , ... , ( n - 1 ) } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices, a closed walk is represented by the following sequence: p(0) e(f(0)) p(1) e(f(1)) ... p(n-1) e(f(n-1)) p(n)=p(0).

Notice that by this definition, a single vertex can be considered as a closed walk of length 0, see also 0clwlk 29380. (Contributed by Alexander van der Vekens, 12-Mar-2018.) (Revised by AV, 16-Feb-2021.)

Assertion

Ref	Expression
df-clwlks	⊢ ClWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})

Distinct variable group:   𝑓,𝑔,𝑝

Detailed syntax breakdown of Definition df-clwlks

Step	Hyp	Ref	Expression
1		cclwlks 29024	. 2 class ClWalks
2		vg	. . 3 setvar 𝑔
3		cvv 3474	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1540	. . . . . 6 class 𝑓
6		vp	. . . . . . 7 setvar 𝑝
7	6	cv 1540	. . . . . 6 class 𝑝
8	2	cv 1540	. . . . . . 7 class 𝑔
9		cwlks 28850	. . . . . . 7 class Walks
10	8, 9	cfv 6543	. . . . . 6 class (Walks‘𝑔)
11	5, 7, 10	wbr 5148	. . . . 5 wff 𝑓(Walks‘𝑔)𝑝
12		cc0 11109	. . . . . . 7 class 0
13	12, 7	cfv 6543	. . . . . 6 class (𝑝‘0)
14		chash 14289	. . . . . . . 8 class ♯
15	5, 14	cfv 6543	. . . . . . 7 class (♯‘𝑓)
16	15, 7	cfv 6543	. . . . . 6 class (𝑝‘(♯‘𝑓))
17	13, 16	wceq 1541	. . . . 5 wff (𝑝‘0) = (𝑝‘(♯‘𝑓))
18	11, 17	wa 396	. . . 4 wff (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))
19	18, 4, 6	copab 5210	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))}
20	2, 3, 19	cmpt 5231	. 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})
21	1, 20	wceq 1541	1 wff ClWalks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓(Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(♯‘𝑓)))})

This definition is referenced by:  clwlks  29026