Definition df-spths

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

According to Wikipedia ("Path (graph theory)", https://en.wikipedia.org/wiki/Path_(graph_theory) , 3-Oct-2017): "A path is a trail in which all vertices (except possibly the first and last) are distinct. ... use the term simple path to refer to a path which contains no repeated vertices."

Therefore, a simple path can be represented by an injective mapping f from { 1 , ... , n } and an injective mapping p from { 0 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the simple path 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, 20-Oct-2017) (Revised by AV, 9-Jan-2021)

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

Detailed syntax breakdown

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