This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Definition df-clwwlkn

Description: Define the set of all closed walks of a fixed length n as words over the set of vertices in a graph g . If 0 < n , 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-clwlks . For n = 0 , the set is empty, see clwwlkn0 . (Contributed by Alexander van der Vekens, 20-Mar-2018) (Revised by AV, 24-Apr-2021) (Revised by AV, 22-Mar-2022)

Ref Expression
Assertion df-clwwlkn ⊢ ClWWalksN = n ∈ ℕ 0 , g ∈ V ⟼ w ∈ ClWWalks ⁡ g | w = n

Detailed syntax breakdown

Step Hyp Ref Expression
0 cclwwlkn class ClWWalksN
1 vn setvar n
2 cn0 class ℕ 0
3 vg setvar g
4 cvv class V
5 vw setvar w
6 cclwwlk class ClWWalks
7 3 cv setvar g
8 7 6 cfv class ClWWalks ⁡ g
9 chash class .
10 5 cv setvar w
11 10 9 cfv class w
12 1 cv setvar n
13 11 12 wceq wff w = n
14 13 5 8 crab class w ∈ ClWWalks ⁡ g | w = n
15 1 3 2 4 14 cmpo class n ∈ ℕ 0 , g ∈ V ⟼ w ∈ ClWWalks ⁡ g | w = n
16 0 15 wceq wff ClWWalksN = n ∈ ℕ 0 , g ∈ V ⟼ w ∈ ClWWalks ⁡ g | w = n