Definition df-wlks 27375

Description: Define the set of all walks (in a hypergraph). Such walks correspond to the s-walks "on the vertex level" (with s = 1), and also to 1-walks "on the edge level" (see wlk1walk 27414) discussed in Aksoy et al. The predicate 𝐹(Walks‘𝐺)𝑃 can be read as "The pair ⟨𝐹, 𝑃⟩ represents a walk in a graph 𝐺", see also iswlk 27386.

The condition {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘)) (hereinafter referred to as C) would not be sufficient, because the repetition of a vertex in a walk (i.e. (𝑝‘𝑘) = (𝑝‘(𝑘 + 1)) should be allowed only if there is a loop at (𝑝‘𝑘). Otherwise, C would be fulfilled by each edge containing (𝑝‘𝑘).

According to the definition of [Bollobas] p. 4.: "A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) ...", a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is 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 AV, 30-Dec-2020.)

Assertion

Ref	Expression
df-wlks	⊢ Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))})

Distinct variable group:   𝑓,𝑔,𝑘,𝑝

Detailed syntax breakdown of Definition df-wlks

Step	Hyp	Ref	Expression
1		cwlks 27372	. 2 class Walks
2		vg	. . 3 setvar 𝑔
3		cvv 3494	. . 3 class V
4		vf	. . . . . . 7 setvar 𝑓
5	4	cv 1532	. . . . . 6 class 𝑓
6	2	cv 1532	. . . . . . . . 9 class 𝑔
7		ciedg 26776	. . . . . . . . 9 class iEdg
8	6, 7	cfv 6349	. . . . . . . 8 class (iEdg‘𝑔)
9	8	cdm 5549	. . . . . . 7 class dom (iEdg‘𝑔)
10	9	cword 13855	. . . . . 6 class Word dom (iEdg‘𝑔)
11	5, 10	wcel 2110	. . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12		cc0 10531	. . . . . . 7 class 0
13		chash 13684	. . . . . . . 8 class ♯
14	5, 13	cfv 6349	. . . . . . 7 class (♯‘𝑓)
15		cfz 12886	. . . . . . 7 class ...
16	12, 14, 15	co 7150	. . . . . 6 class (0...(♯‘𝑓))
17		cvtx 26775	. . . . . . 7 class Vtx
18	6, 17	cfv 6349	. . . . . 6 class (Vtx‘𝑔)
19		vp	. . . . . . 7 setvar 𝑝
20	19	cv 1532	. . . . . 6 class 𝑝
21	16, 18, 20	wf 6345	. . . . 5 wff 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔)
22		vk	. . . . . . . . . 10 setvar 𝑘
23	22	cv 1532	. . . . . . . . 9 class 𝑘
24	23, 20	cfv 6349	. . . . . . . 8 class (𝑝‘𝑘)
25		c1 10532	. . . . . . . . . 10 class 1
26		caddc 10534	. . . . . . . . . 10 class +
27	23, 25, 26	co 7150	. . . . . . . . 9 class (𝑘 + 1)
28	27, 20	cfv 6349	. . . . . . . 8 class (𝑝‘(𝑘 + 1))
29	24, 28	wceq 1533	. . . . . . 7 wff (𝑝‘𝑘) = (𝑝‘(𝑘 + 1))
30	23, 5	cfv 6349	. . . . . . . . 9 class (𝑓‘𝑘)
31	30, 8	cfv 6349	. . . . . . . 8 class ((iEdg‘𝑔)‘(𝑓‘𝑘))
32	24	csn 4560	. . . . . . . 8 class {(𝑝‘𝑘)}
33	31, 32	wceq 1533	. . . . . . 7 wff ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}
34	24, 28	cpr 4562	. . . . . . . 8 class {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}
35	34, 31	wss 3935	. . . . . . 7 wff {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))
36	29, 33, 35	wif 1057	. . . . . 6 wff if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘)))
37		cfzo 13027	. . . . . . 7 class ..^
38	12, 14, 37	co 7150	. . . . . 6 class (0..^(♯‘𝑓))
39	36, 22, 38	wral 3138	. . . . 5 wff ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘)))
40	11, 21, 39	w3a 1083	. . . 4 wff (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))
41	40, 4, 19	copab 5120	. . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))}
42	2, 3, 41	cmpt 5138	. 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))})
43	1, 42	wceq 1533	1 wff Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(♯‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(♯‘𝑓))if-((𝑝‘𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓‘𝑘)) = {(𝑝‘𝑘)}, {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓‘𝑘))))})

This definition is referenced by:  wksfval  27385  relwlk  27401