Mathbox for Alexander van der Vekens < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  df-wwlksn Structured version   Visualization version   GIF version

Definition df-wwlksn 41034
 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-1wlks 40800. (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 41029 . 2 class WWalkSN
2 vn . . 3 setvar 𝑛
3 vg . . 3 setvar 𝑔
4 cn0 11169 . . 3 class 0
5 cvv 3173 . . 3 class V
6 vw . . . . . . 7 setvar 𝑤
76cv 1474 . . . . . 6 class 𝑤
8 chash 12979 . . . . . 6 class #
97, 8cfv 5804 . . . . 5 class (#‘𝑤)
102cv 1474 . . . . . 6 class 𝑛
11 c1 9816 . . . . . 6 class 1
12 caddc 9818 . . . . . 6 class +
1310, 11, 12co 6549 . . . . 5 class (𝑛 + 1)
149, 13wceq 1475 . . . 4 wff (#‘𝑤) = (𝑛 + 1)
153cv 1474 . . . . 5 class 𝑔
16 cwwlks 41028 . . . . 5 class WWalkS
1715, 16cfv 5804 . . . 4 class (WWalkS‘𝑔)
1814, 6, 17crab 2900 . . 3 class {𝑤 ∈ (WWalkS‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)}
192, 3, 4, 5, 18cmpt2 6551 . 2 class (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalkS‘𝑔) ∣ (#‘𝑤) = (𝑛 + 1)})
201, 19wceq 1475 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  41040  wwlknbp  41044  wspthsn  41046  iswwlksnon  41051
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