df-upwlks

Description: Define the set of all walks (in a pseudograph), called "simple walks" in the following.

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory) , 3-Oct-2017): "A walk of length k in a graph is an alternating sequence of vertices and edges, v0 , e0 , v1 , e1 , v2 , ... , v(k-1) , e(k-1) , v(k) which begins and ends with vertices. If the graph is undirected, then the endpoints of e(i) are v(i) and v(i+1)."

According to Bollobas: " A walk W in a graph is an alternating sequence of vertices and edges x0 , e1 , x1 , e2 , ... , e(l) , x(l) where e(i) = x(i-1)x(i), 0Bollobas p. 4.

Therefore, a walk can be represented by two mappings f from { 1 , ... , n } and p from { 0 , ... , n }, where f enumerates the (indices of the) edges, and p enumerates the vertices. So the walk is represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n).

Although this definition is also applicable for arbitrary hypergraphs, it allows only walks consisting of not proper hyperedges (i.e. edges connecting at most two vertices). Therefore, it should be used for pseudographs only. (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017) (Revised by AV, 28-Dec-2020)

		Ref	Expression
	Assertion	df-upwlks	$⊢ UPWalks = (g \in V ⟼ \{⟨f, p⟩ \| (f \in Word dom iEdg (g) \land p : (0 \dots \|f\|) ⟶ Vtx (g) \land \forall k \in (0 ..^\|f\|) iEdg (g) (f (k)) = \{p (k), p (k + 1)\})\})$

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cupwlks	$class UPWalks$
1		vg	$setvar g$
2		cvv	$class V$
3		vf	$setvar f$
4		vp	$setvar p$
5	3	cv	$setvar f$
6		ciedg	$class iEdg$
7	1	cv	$setvar g$
8	7 6	cfv	$class iEdg (g)$
9	8	cdm	$class dom iEdg (g)$
10	9	cword	$class Word dom iEdg (g)$
11	5 10	wcel	$wff f \in Word dom iEdg (g)$
12	4	cv	$setvar p$
13		cc0	$class 0$
14		cfz	$class \dots$
15		chash	$class \|.\|$
16	5 15	cfv	$class \|f\|$
17	13 16 14	co	$class (0 \dots \|f\|)$
18		cvtx	$class Vtx$
19	7 18	cfv	$class Vtx (g)$
20	17 19 12	wf	$wff p : (0 \dots \|f\|) ⟶ Vtx (g)$
21		vk	$setvar k$
22		cfzo	$class ..^$
23	13 16 22	co	$class (0 ..^\|f\|)$
24	21	cv	$setvar k$
25	24 5	cfv	$class f (k)$
26	25 8	cfv	$class iEdg (g) (f (k))$
27	24 12	cfv	$class p (k)$
28		caddc	$class +$
29		c1	$class 1$
30	24 29 28	co	$class (k + 1)$
31	30 12	cfv	$class p (k + 1)$
32	27 31	cpr	$class \{p (k), p (k + 1)\}$
33	26 32	wceq	$wff iEdg (g) (f (k)) = \{p (k), p (k + 1)\}$
34	33 21 23	wral	$wff \forall k \in (0 ..^\|f\|) iEdg (g) (f (k)) = \{p (k), p (k + 1)\}$
35	11 20 34	w3a	$wff (f \in Word dom iEdg (g) \land p : (0 \dots \|f\|) ⟶ Vtx (g) \land \forall k \in (0 ..^\|f\|) iEdg (g) (f (k)) = \{p (k), p (k + 1)\})$
36	35 3 4	copab	$class \{⟨f, p⟩ \| (f \in Word dom iEdg (g) \land p : (0 \dots \|f\|) ⟶ Vtx (g) \land \forall k \in (0 ..^\|f\|) iEdg (g) (f (k)) = \{p (k), p (k + 1)\})\}$
37	1 2 36	cmpt	$class (g \in V ⟼ \{⟨f, p⟩ \| (f \in Word dom iEdg (g) \land p : (0 \dots \|f\|) ⟶ Vtx (g) \land \forall k \in (0 ..^\|f\|) iEdg (g) (f (k)) = \{p (k), p (k + 1)\})\})$
38	0 37	wceq	$wff UPWalks = (g \in V ⟼ \{⟨f, p⟩ \| (f \in Word dom iEdg (g) \land p : (0 \dots \|f\|) ⟶ Vtx (g) \land \forall k \in (0 ..^\|f\|) iEdg (g) (f (k)) = \{p (k), p (k + 1)\})\})$

Metamath Proof Explorer

Definition df-upwlks

Detailed syntax breakdown