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Definition df-1wlks 40795
Description: Define the set of all 1-walks (in a hypergraph). Such 1-walks correspond to the s-walks "on the vertex level" (with s = 1), and also to 1-walks "on the edge level" (see 1wlk1walk 40838) discussed in Aksoy et al. The predicate 𝐹(1Walks‘𝐺)𝑃 can be read as "The pair 𝐹, 𝑃 represents a walk in a graph 𝐺", see also is1wlk 40808.

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

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-1wlks 1Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))))})
Distinct variable group:   𝑓,𝑔,𝑘,𝑝

Detailed syntax breakdown of Definition df-1wlks
StepHypRef Expression
1 c1wlks 40791 . 2 class 1Walks
2 vg . . 3 setvar 𝑔
3 cvv 3172 . . 3 class V
4 vf . . . . . . 7 setvar 𝑓
54cv 1473 . . . . . 6 class 𝑓
62cv 1473 . . . . . . . . 9 class 𝑔
7 ciedg 40225 . . . . . . . . 9 class iEdg
86, 7cfv 5790 . . . . . . . 8 class (iEdg‘𝑔)
98cdm 5028 . . . . . . 7 class dom (iEdg‘𝑔)
109cword 13092 . . . . . 6 class Word dom (iEdg‘𝑔)
115, 10wcel 1976 . . . . 5 wff 𝑓 ∈ Word dom (iEdg‘𝑔)
12 cc0 9792 . . . . . . 7 class 0
13 chash 12934 . . . . . . . 8 class #
145, 13cfv 5790 . . . . . . 7 class (#‘𝑓)
15 cfz 12152 . . . . . . 7 class ...
1612, 14, 15co 6527 . . . . . 6 class (0...(#‘𝑓))
17 cvtx 40224 . . . . . . 7 class Vtx
186, 17cfv 5790 . . . . . 6 class (Vtx‘𝑔)
19 vp . . . . . . 7 setvar 𝑝
2019cv 1473 . . . . . 6 class 𝑝
2116, 18, 20wf 5786 . . . . 5 wff 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔)
22 vk . . . . . . . . . 10 setvar 𝑘
2322cv 1473 . . . . . . . . 9 class 𝑘
2423, 20cfv 5790 . . . . . . . 8 class (𝑝𝑘)
25 c1 9793 . . . . . . . . . 10 class 1
26 caddc 9795 . . . . . . . . . 10 class +
2723, 25, 26co 6527 . . . . . . . . 9 class (𝑘 + 1)
2827, 20cfv 5790 . . . . . . . 8 class (𝑝‘(𝑘 + 1))
2924, 28wceq 1474 . . . . . . 7 wff (𝑝𝑘) = (𝑝‘(𝑘 + 1))
3023, 5cfv 5790 . . . . . . . . 9 class (𝑓𝑘)
3130, 8cfv 5790 . . . . . . . 8 class ((iEdg‘𝑔)‘(𝑓𝑘))
3224csn 4124 . . . . . . . 8 class {(𝑝𝑘)}
3331, 32wceq 1474 . . . . . . 7 wff ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}
3424, 28cpr 4126 . . . . . . . 8 class {(𝑝𝑘), (𝑝‘(𝑘 + 1))}
3534, 31wss 3539 . . . . . . 7 wff {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))
3629, 33, 35wif 1005 . . . . . 6 wff if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘)))
37 cfzo 12289 . . . . . . 7 class ..^
3812, 14, 37co 6527 . . . . . 6 class (0..^(#‘𝑓))
3936, 22, 38wral 2895 . . . . 5 wff 𝑘 ∈ (0..^(#‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘)))
4011, 21, 39w3a 1030 . . . 4 wff (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))))
4140, 4, 19copab 4636 . . 3 class {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))))}
422, 3, 41cmpt 4637 . 2 class (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))))})
431, 42wceq 1474 1 wff 1Walks = (𝑔 ∈ V ↦ {⟨𝑓, 𝑝⟩ ∣ (𝑓 ∈ Word dom (iEdg‘𝑔) ∧ 𝑝:(0...(#‘𝑓))⟶(Vtx‘𝑔) ∧ ∀𝑘 ∈ (0..^(#‘𝑓))if-((𝑝𝑘) = (𝑝‘(𝑘 + 1)), ((iEdg‘𝑔)‘(𝑓𝑘)) = {(𝑝𝑘)}, {(𝑝𝑘), (𝑝‘(𝑘 + 1))} ⊆ ((iEdg‘𝑔)‘(𝑓𝑘))))})
Colors of variables: wff setvar class
This definition is referenced by:  1wlksfval  40806  rel1wlk  40825
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