Definition df-trls

Description: Define the set of all Trails (in an undirected graph).

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory) , 3-Oct-2017): "A trail is a walk in which all edges are distinct.

According to Bollobas: "... walk is called a trail if all its edges are distinct.", see Definition of Bollobas p. 5.

Therefore, a trail can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the trail is also represented by the following sequence: p(0) e(f(1)) p(1) e(f(2)) ... p(n-1) e(f(n)) p(n). (Contributed by Alexander van der Vekens and Mario Carneiro, 4-Oct-2017) (Revised by AV, 28-Dec-2020)

	Ref	Expression
Assertion	df-trls	|- Trails = ( g e. _V |-> { <. f , p >. | ( f ( Walks ` g ) p /\ Fun `' f ) } )

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		ctrls	|- Trails
1		vg	|- g
2		cvv	|- _V
3		vf	|- f
4		vp	|- p
5	3	cv	|- f
6		cwlks	|- Walks
7	1	cv	|- g
8	7 6	cfv	|- ( Walks ` g )
9	4	cv	|- p
10	5 9 8	wbr	|- f ( Walks ` g ) p
11	5	ccnv	|- `' f
12	11	wfun	|- Fun `' f
13	10 12	wa	|- ( f ( Walks ` g ) p /\ Fun `' f )
14	13 3 4	copab	|- { <. f , p >. | ( f ( Walks ` g ) p /\ Fun `' f ) }
15	1 2 14	cmpt	|- ( g e. _V |-> { <. f , p >. | ( f ( Walks ` g ) p /\ Fun `' f ) } )
16	0 15	wceq	|- Trails = ( g e. _V |-> { <. f , p >. | ( f ( Walks ` g ) p /\ Fun `' f ) } )