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Definition df-clwwlkn 26280
 Description: Define the set of all Closed 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-1) of the vertices in a closed walk p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n)=p(0) as defined in df-clwlk 26278. (Contributed by Alexander van der Vekens, 20-Mar-2018.)
Assertion
Ref Expression
df-clwwlkn ClWWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}))
Distinct variable group:   𝑒,𝑛,𝑣,𝑤

Detailed syntax breakdown of Definition df-clwwlkn
StepHypRef Expression
1 cclwwlkn 26277 . 2 class ClWWalksN
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 . . . . . 6 class 𝑛
1210, 11wceq 1475 . . . . 5 wff (#‘𝑤) = 𝑛
132cv 1474 . . . . . 6 class 𝑣
143cv 1474 . . . . . 6 class 𝑒
15 cclwwlk 26276 . . . . . 6 class ClWWalks
1613, 14, 15co 6549 . . . . 5 class (𝑣 ClWWalks 𝑒)
1712, 7, 16crab 2900 . . . 4 class {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}
185, 6, 17cmpt 4643 . . 3 class (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛})
192, 3, 4, 4, 18cmpt2 6551 . 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}))
201, 19wceq 1475 1 wff ClWWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 ClWWalks 𝑒) ∣ (#‘𝑤) = 𝑛}))
 Colors of variables: wff setvar class This definition is referenced by:  clwwlkn  26295  clwwlknprop  26300
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