Definition df-pths 26612

Description: Define the set of all 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."

According to Bollobas: "... a path is a walk with distinct vertices.", see Notation of [Bollobas] p. 5. (A walk with distinct vertices is actually a simple path, see upgrwlkdvspth 26635).

Therefore, a path can be represented by an injective mapping f from { 1 , ... , n } and a mapping p from { 0 , ... , n }, which is injective restricted to the set { 1 , ... , n }, where f enumerates the (indices of the) different edges, and p enumerates the vertices. So the 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 and Mario Carneiro, 4-Oct-2017.) (Revised by AV, 9-Jan-2021.)

Assertion

Ref	Expression
df-pths	|- Paths = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ Fun `' ( p |` ( 1..^ ( # ` f ) ) ) /\ ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1..^ ( # ` f ) ) ) ) = (/) ) } )

Distinct variable group:   f, g, p

Detailed syntax breakdown of Definition df-pths

Step	Hyp	Ref	Expression
1		cpths 26608	. 2 class Paths
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		c1 9937	. . . . . . . . 9 class 1
13		chash 13117	. . . . . . . . . 10 class #
14	5, 13	cfv 5888	. . . . . . . . 9 class ( # ` f )
15		cfzo 12465	. . . . . . . . 9 class ..^
16	12, 14, 15	co 6650	. . . . . . . 8 class ( 1..^ ( # ` f ) )
17	7, 16	cres 5116	. . . . . . 7 class ( p |` ( 1..^ ( # ` f ) ) )
18	17	ccnv 5113	. . . . . 6 class `' ( p |` ( 1..^ ( # ` f ) ) )
19	18	wfun 5882	. . . . 5 wff Fun `' ( p |` ( 1..^ ( # ` f ) ) )
20		cc0 9936	. . . . . . . . 9 class 0
21	20, 14	cpr 4179	. . . . . . . 8 class { 0 , ( # ` f ) }
22	7, 21	cima 5117	. . . . . . 7 class ( p " { 0 , ( # ` f ) } )
23	7, 16	cima 5117	. . . . . . 7 class ( p " ( 1..^ ( # ` f ) ) )
24	22, 23	cin 3573	. . . . . 6 class ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1..^ ( # ` f ) ) ) )
25		c0 3915	. . . . . 6 class (/)
26	24, 25	wceq 1483	. . . . 5 wff ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1..^ ( # ` f ) ) ) ) = (/)
27	11, 19, 26	w3a 1037	. . . 4 wff ( f (Trails ` g ) p /\ Fun `' ( p |` ( 1..^ ( # ` f ) ) ) /\ ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1..^ ( # ` f ) ) ) ) = (/) )
28	27, 4, 6	copab 4712	. . 3 class { <. f , p >. | ( f (Trails ` g ) p /\ Fun `' ( p |` ( 1..^ ( # ` f ) ) ) /\ ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1..^ ( # ` f ) ) ) ) = (/) ) }
29	2, 3, 28	cmpt 4729	. 2 class ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ Fun `' ( p |` ( 1..^ ( # ` f ) ) ) /\ ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1..^ ( # ` f ) ) ) ) = (/) ) } )
30	1, 29	wceq 1483	1 wff Paths = ( g e. _V |-> { <. f , p >. | ( f (Trails ` g ) p /\ Fun `' ( p |` ( 1..^ ( # ` f ) ) ) /\ ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1..^ ( # ` f ) ) ) ) = (/) ) } )

This definition is referenced by:  relpths  26616  pthsfval  26617