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

Step	Hyp	Ref	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 ℕ0
7		vw	. . . . . . . 8 setvar 𝑤
8	7	cv 1474	. . . . . . 7 class 𝑤
9		chash 12979	. . . . . . 7 class #
10	8, 9	cfv 5804	. . . . . 6 class (#‘𝑤)
11	5	cv 1474	. . . . . . 7 class 𝑛
12		c1 9816	. . . . . . 7 class 1
13		caddc 9818	. . . . . . 7 class +
14	11, 12, 13	co 6549	. . . . . 6 class (𝑛 + 1)
15	10, 14	wceq 1475	. . . . 5 wff (#‘𝑤) = (𝑛 + 1)
16	2	cv 1474	. . . . . 6 class 𝑣
17	3	cv 1474	. . . . . 6 class 𝑒
18		cwwlk 26205	. . . . . 6 class WWalks
19	16, 17, 18	co 6549	. . . . 5 class (𝑣 WWalks 𝑒)
20	15, 7, 19	crab 2900	. . . 4 class {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)}
21	5, 6, 20	cmpt 4643	. . 3 class (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)})
22	2, 3, 4, 4, 21	cmpt2 6551	. 2 class (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)}))
23	1, 22	wceq 1475	1 wff WWalksN = (𝑣 ∈ V, 𝑒 ∈ V ↦ (𝑛 ∈ ℕ0 ↦ {𝑤 ∈ (𝑣 WWalks 𝑒) ∣ (#‘𝑤) = (𝑛 + 1)}))

This definition is referenced by:  wwlkn  26210  wwlknprop  26214