MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-wwlksn Structured version   Visualization version   GIF version

Definition df-wwlksn 26855
Description: Define the set of all walks (in an undirected graph) of a fixed length n as words over the set of vertices. Such a word corresponds to the sequence p(0) p(1) ... p(n) of the vertices in a walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n) as defined in df-wlks 26626. (Contributed by Alexander van der Vekens, 15-Jul-2018.) (Revised by AV, 8-Apr-2021.)
Assertion
Ref Expression
df-wwlksn WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
Distinct variable group:   𝑔,𝑛,𝑤

Detailed syntax breakdown of Definition df-wwlksn
StepHypRef Expression
1 cwwlksn 26850 . 2 class WWalksN
2 vn . . 3 setvar 𝑛
3 vg . . 3 setvar 𝑔
4 cn0 11405 . . 3 class 0
5 cvv 3304 . . 3 class V
6 vw . . . . . . 7 setvar 𝑤
76cv 1595 . . . . . 6 class 𝑤
8 chash 13232 . . . . . 6 class
97, 8cfv 6001 . . . . 5 class (♯‘𝑤)
102cv 1595 . . . . . 6 class 𝑛
11 c1 10050 . . . . . 6 class 1
12 caddc 10052 . . . . . 6 class +
1310, 11, 12co 6765 . . . . 5 class (𝑛 + 1)
149, 13wceq 1596 . . . 4 wff (♯‘𝑤) = (𝑛 + 1)
153cv 1595 . . . . 5 class 𝑔
16 cwwlks 26849 . . . . 5 class WWalks
1715, 16cfv 6001 . . . 4 class (WWalks‘𝑔)
1814, 6, 17crab 3018 . . 3 class {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)}
192, 3, 4, 5, 18cmpt2 6767 . 2 class (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
201, 19wceq 1596 1 wff WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
Colors of variables: wff setvar class
This definition is referenced by:  wwlksn  26861  wwlknbp  26866  wspthsn  26873  iswwlksnon  26878  iswwlksnonOLD  26879
  Copyright terms: Public domain W3C validator