Definition df-spths 26613

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.)

Assertion

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

Distinct variable group:   f, g, p

Detailed syntax breakdown of Definition df-spths

Step	Hyp	Ref	Expression
1		cspths 26609	. 2 class SPaths
2		vg	. . 3 setvar g
3		cvv 3200	. . 3 class _V
4		vf	. . . . . . 7 setvar f
5	4	cv 1482	. . . . . 6 class f
6		vp	. . . . . . 7 setvar p
7	6	cv 1482	. . . . . 6 class p
8	2	cv 1482	. . . . . . 7 class g
9		ctrls 26587	. . . . . . 7 class Trails
10	8, 9	cfv 5888	. . . . . 6 class (Trails ` g )
11	5, 7, 10	wbr 4653	. . . . 5 wff f (Trails ` g ) p
12	7	ccnv 5113	. . . . . 6 class `' p
13	12	wfun 5882	. . . . 5 wff Fun `' p
14	11, 13	wa 384	. . . 4 wff ( f (Trails ` g ) p /\ Fun `' p )
15	14, 4, 6	copab 4712	. . 3 class { <. f , p >. | ( f (Trails ` g ) p /\ Fun `' p ) }
16	2, 3, 15	cmpt 4729	. 2 class ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ Fun `' p ) } )
17	1, 16	wceq 1483	1 wff SPaths = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ Fun `' p ) } )

This definition is referenced by:  spthsfval  26618