Definition df-wwlksn 30420

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-wlks 30180. (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

Step	Hyp	Ref	Expression
1		cwwlksn 30415	. 2 class WWalksN
2		vn	. . 3 setvar 𝑛
3		vg	. . 3 setvar 𝑔
4		cn0 12606	. . 3 class ℕ0
5		cvv 3451	. . 3 class V
6		vw	. . . . . . 7 setvar 𝑤
7	6	cv 1569	. . . . . 6 class 𝑤
8		chash 14474	. . . . . 6 class ♯
9	7, 8	cfv 6538	. . . . 5 class (♯‘𝑤)
10	2	cv 1569	. . . . . 6 class 𝑛
11		c1 11201	. . . . . 6 class 1
12		caddc 11203	. . . . . 6 class +
13	10, 11, 12	co 7420	. . . . 5 class (𝑛 + 1)
14	9, 13	wceq 1570	. . . 4 wff (♯‘𝑤) = (𝑛 + 1)
15	3	cv 1569	. . . . 5 class 𝑔
16		cwwlks 30414	. . . . 5 class WWalks
17	15, 16	cfv 6538	. . . 4 class (WWalks‘𝑔)
18	14, 6, 17	crab 3413	. . 3 class {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)}
19	2, 3, 4, 5, 18	cmpo 7422	. 2 class (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})
20	1, 19	wceq 1570	1 wff WWalksN = (𝑛 ∈ ℕ0, 𝑔 ∈ V ↦ {𝑤 ∈ (WWalks‘𝑔) ∣ (♯‘𝑤) = (𝑛 + 1)})

This definition is used by:  wwlksn  30426  wwlknbp  30431  wspthsn  30437  iswwlksnon  30442