Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-wlk | Structured version Visualization version GIF version |
Description: Define the set of all
Walks (in an undirected graph).
According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory), 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)." According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0<i<=l.", see Definition of [Bollobas] p. 4. Therefore, 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 Alexander van der Vekens and Mario Carneiro, 4-Oct-2017.) |
Ref | Expression |
---|---|
df-wlk | ⊢ Walks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cwalk 26026 | . 2 class Walks | |
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 | 3 | cv 1474 | . . . . . . . 8 class 𝑒 |
8 | 7 | cdm 5038 | . . . . . . 7 class dom 𝑒 |
9 | 8 | cword 13146 | . . . . . 6 class Word dom 𝑒 |
10 | 6, 9 | wcel 1977 | . . . . 5 wff 𝑓 ∈ Word dom 𝑒 |
11 | cc0 9815 | . . . . . . 7 class 0 | |
12 | chash 12979 | . . . . . . . 8 class # | |
13 | 6, 12 | cfv 5804 | . . . . . . 7 class (#‘𝑓) |
14 | cfz 12197 | . . . . . . 7 class ... | |
15 | 11, 13, 14 | co 6549 | . . . . . 6 class (0...(#‘𝑓)) |
16 | 2 | cv 1474 | . . . . . 6 class 𝑣 |
17 | vp | . . . . . . 7 setvar 𝑝 | |
18 | 17 | cv 1474 | . . . . . 6 class 𝑝 |
19 | 15, 16, 18 | wf 5800 | . . . . 5 wff 𝑝:(0...(#‘𝑓))⟶𝑣 |
20 | vk | . . . . . . . . . 10 setvar 𝑘 | |
21 | 20 | cv 1474 | . . . . . . . . 9 class 𝑘 |
22 | 21, 6 | cfv 5804 | . . . . . . . 8 class (𝑓‘𝑘) |
23 | 22, 7 | cfv 5804 | . . . . . . 7 class (𝑒‘(𝑓‘𝑘)) |
24 | 21, 18 | cfv 5804 | . . . . . . . 8 class (𝑝‘𝑘) |
25 | c1 9816 | . . . . . . . . . 10 class 1 | |
26 | caddc 9818 | . . . . . . . . . 10 class + | |
27 | 21, 25, 26 | co 6549 | . . . . . . . . 9 class (𝑘 + 1) |
28 | 27, 18 | cfv 5804 | . . . . . . . 8 class (𝑝‘(𝑘 + 1)) |
29 | 24, 28 | cpr 4127 | . . . . . . 7 class {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} |
30 | 23, 29 | wceq 1475 | . . . . . 6 wff (𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} |
31 | cfzo 12334 | . . . . . . 7 class ..^ | |
32 | 11, 13, 31 | co 6549 | . . . . . 6 class (0..^(#‘𝑓)) |
33 | 30, 20, 32 | wral 2896 | . . . . 5 wff ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))} |
34 | 10, 19, 33 | w3a 1031 | . . . 4 wff (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))}) |
35 | 34, 5, 17 | copab 4642 | . . 3 class {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})} |
36 | 2, 3, 4, 4, 35 | cmpt2 6551 | . 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) |
37 | 1, 36 | wceq 1475 | 1 wff Walks = (𝑣 ∈ V, 𝑒 ∈ V ↦ {〈𝑓, 𝑝〉 ∣ (𝑓 ∈ Word dom 𝑒 ∧ 𝑝:(0...(#‘𝑓))⟶𝑣 ∧ ∀𝑘 ∈ (0..^(#‘𝑓))(𝑒‘(𝑓‘𝑘)) = {(𝑝‘𝑘), (𝑝‘(𝑘 + 1))})}) |
Colors of variables: wff setvar class |
This definition is referenced by: relwlk 26046 wlks 26047 2mwlk 26049 wlkbprop 26051 wlkcompim 26054 wlkelwrd 26058 wlkdvspth 26138 |
Copyright terms: Public domain | W3C validator |