Definition df-clwwlkn 30044

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 29791. For 𝑛 = 0, the set is empty, see clwwlkn0 30047. (Contributed by Alexander van der Vekens, 20-Mar-2018.) (Revised by AV, 24-Apr-2021.) (Revised by AV, 22-Mar-2022.)

Assertion

Ref	Expression
df-clwwlkn	⊢ ClWWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛})

Distinct variable group:   𝑔,𝑛,𝑤

Detailed syntax breakdown of Definition df-clwwlkn

Step	Hyp	Ref	Expression
1		cclwwlkn 30043	. 2 class ClWWalksN
2		vn	. . 3 setvar 𝑛
3		vg	. . 3 setvar 𝑔
4		cn0 12526	. . 3 class ℕ0
5		cvv 3480	. . 3 class V
6		vw	. . . . . . 7 setvar 𝑤
7	6	cv 1539	. . . . . 6 class 𝑤
8		chash 14369	. . . . . 6 class ♯
9	7, 8	cfv 6561	. . . . 5 class (♯‘𝑤)
10	2	cv 1539	. . . . 5 class 𝑛
11	9, 10	wceq 1540	. . . 4 wff (♯‘𝑤) = 𝑛
12	3	cv 1539	. . . . 5 class 𝑔
13		cclwwlk 30000	. . . . 5 class ClWWalks
14	12, 13	cfv 6561	. . . 4 class (ClWWalks‘𝑔)
15	11, 6, 14	crab 3436	. . 3 class {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛}
16	2, 3, 4, 5, 15	cmpo 7433	. 2 class (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛})
17	1, 16	wceq 1540	1 wff ClWWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (ClWWalks‘𝑔) ∣ (♯‘𝑤) = 𝑛})

This definition is referenced by:  clwwlkn  30045  clwwlkneq0  30048