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Definition df-wwlkn 26208
 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-wlk 26036. (Contributed by Alexander van der Vekens, 15-Jul-2018.)
Assertion
Ref Expression
df-wwlkn WWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)}))
Distinct variable group:   𝑒,𝑛,𝑣,𝑤

Detailed syntax breakdown of Definition df-wwlkn
StepHypRef Expression
1 cwwlkn 26206 . 2 class WWalksN
2 vv . . 3 setvar 𝑣
3 ve . . 3 setvar 𝑒
4 cvv 3173 . . 3 class V
5 vn . . . 4 setvar 𝑛
6 cn0 11169 . . . 4 class 0
7 vw . . . . . . . 8 setvar 𝑤
87cv 1474 . . . . . . 7 class 𝑤
9 chash 12979 . . . . . . 7 class #
108, 9cfv 5804 . . . . . 6 class (#‘𝑤)
115cv 1474 . . . . . . 7 class 𝑛
12 c1 9816 . . . . . . 7 class 1
13 caddc 9818 . . . . . . 7 class +
1411, 12, 13co 6549 . . . . . 6 class (𝑛 + 1)
1510, 14wceq 1475 . . . . 5 wff (#‘𝑤) = (𝑛 + 1)
162cv 1474 . . . . . 6 class 𝑣
173cv 1474 . . . . . 6 class 𝑒
18 cwwlk 26205 . . . . . 6 class WWalks
1916, 17, 18co 6549 . . . . 5 class (𝑣 WWalks 𝑒)
2015, 7, 19crab 2900 . . . 4 class {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)}
215, 6, 20cmpt 4643 . . 3 class (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)})
222, 3, 4, 4, 21cmpt2 6551 . 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)}))
231, 22wceq 1475 1 wff WWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)}))
 Colors of variables: wff setvar class This definition is referenced by:  wwlkn  26210  wwlknprop  26214
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