Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-clwlk | 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 { 1 , ... , n } 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(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0). Notice that by this definition, a single vertex is a closed walk of length 0, see also 0clwlk 26293! (Contributed by Alexander van der Vekens, 12-Mar-2018.) |
Ref | Expression |
---|---|
df-clwlk | ⊢ ClWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cclwlk 26275 | . 2 class ClWalks | |
2 | vv | . . 3 setvar 𝑣 | |
3 | ve | . . 3 setvar 𝑒 | |
4 | cvv 3173 | . . 3 class V | |
5 | vf | . . . . . . 7 setvar 𝑓 | |
6 | 5 | cv 1474 | . . . . . 6 class 𝑓 |
7 | vp | . . . . . . 7 setvar 𝑝 | |
8 | 7 | cv 1474 | . . . . . 6 class 𝑝 |
9 | 2 | cv 1474 | . . . . . . 7 class 𝑣 |
10 | 3 | cv 1474 | . . . . . . 7 class 𝑒 |
11 | cwalk 26026 | . . . . . . 7 class Walks | |
12 | 9, 10, 11 | co 6549 | . . . . . 6 class (𝑣 Walks 𝑒) |
13 | 6, 8, 12 | wbr 4583 | . . . . 5 wff 𝑓(𝑣 Walks 𝑒)𝑝 |
14 | cc0 9815 | . . . . . . 7 class 0 | |
15 | 14, 8 | cfv 5804 | . . . . . 6 class (𝑝‘0) |
16 | chash 12979 | . . . . . . . 8 class # | |
17 | 6, 16 | cfv 5804 | . . . . . . 7 class (#‘𝑓) |
18 | 17, 8 | cfv 5804 | . . . . . 6 class (𝑝‘(#‘𝑓)) |
19 | 15, 18 | wceq 1475 | . . . . 5 wff (𝑝‘0) = (𝑝‘(#‘𝑓)) |
20 | 13, 19 | wa 383 | . . . 4 wff (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓))) |
21 | 20, 5, 7 | copab 4642 | . . 3 class {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))} |
22 | 2, 3, 4, 4, 21 | cmpt2 6551 | . 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
23 | 1, 22 | wceq 1475 | 1 wff ClWalks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓(𝑣 Walks 𝑒)𝑝 ∧ (𝑝‘0) = (𝑝‘(#‘𝑓)))}) |
Colors of variables: wff setvar class |
This definition is referenced by: clwlk 26281 isclwlkg 26283 clwlkiswlk 26285 clwlkswlks 26286 clwlkcompim 26292 |
Copyright terms: Public domain | W3C validator |