Mathbox for Alexander van der Vekens |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > df-clwlks | Structured version Visualization version GIF version |
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 is a closed walk of length 0, see also 0clWlk 41298! (Contributed by Alexander van der Vekens, 12-Mar-2018.) (Revised by AV, 16-Feb-2021.) |
Ref | Expression |
---|---|
df-clwlks | ⊢ ClWalkS = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cclwlks 40976 | . 2 class ClWalkS | |
2 | vg | . . 3 setvar 𝑔 | |
3 | cvv 3173 | . . 3 class V | |
4 | vf | . . . . . . 7 setvar 𝑓 | |
5 | 4 | cv 1474 | . . . . . 6 class 𝑓 |
6 | vp | . . . . . . 7 setvar 𝑝 | |
7 | 6 | cv 1474 | . . . . . 6 class 𝑝 |
8 | 2 | cv 1474 | . . . . . . 7 class 𝑔 |
9 | c1wlks 40796 | . . . . . . 7 class 1Walks | |
10 | 8, 9 | cfv 5804 | . . . . . 6 class (1Walks‘𝑔) |
11 | 5, 7, 10 | wbr 4583 | . . . . 5 wff 𝑓(1Walks‘𝑔)𝑝 |
12 | cc0 9815 | . . . . . . 7 class 0 | |
13 | 12, 7 | cfv 5804 | . . . . . 6 class (𝑝‘0) |
14 | chash 12979 | . . . . . . . 8 class # | |
15 | 5, 14 | cfv 5804 | . . . . . . 7 class (#‘𝑓) |
16 | 15, 7 | cfv 5804 | . . . . . 6 class (𝑝‘(#‘𝑓)) |
17 | 13, 16 | wceq 1475 | . . . . 5 wff (𝑝‘0) = (𝑝‘(#‘𝑓)) |
18 | 11, 17 | wa 383 | . . . 4 wff (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓))) |
19 | 18, 4, 6 | copab 4642 | . . 3 class {〈𝑓, 𝑝〉 ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))} |
20 | 2, 3, 19 | cmpt 4643 | . 2 class (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
21 | 1, 20 | wceq 1475 | 1 wff ClWalkS = (𝑔 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(1Walks‘𝑔)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
Colors of variables: wff setvar class |
This definition is referenced by: clwlkS 40978 |
Copyright terms: Public domain | W3C validator |