Definition df-pths

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 ).

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)

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

Detailed syntax breakdown

Step	Hyp	Ref	Expression
0		cpths	|- Paths
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		c1	|- 1
12		cfzo	|- ..^
13		chash	|- #
14	5 13	cfv	|- ( # ` f )
15	11 14 12	co	|- ( 1 ..^ ( # ` f ) )
16	9 15	cres	|- ( p |` ( 1 ..^ ( # ` f ) ) )
17	16	ccnv	|- `' ( p |` ( 1 ..^ ( # ` f ) ) )
18	17	wfun	|- Fun `' ( p |` ( 1 ..^ ( # ` f ) ) )
19		cc0	|- 0
20	19 14	cpr	|- { 0 , ( # ` f ) }
21	9 20	cima	|- ( p " { 0 , ( # ` f ) } )
22	9 15	cima	|- ( p " ( 1 ..^ ( # ` f ) ) )
23	21 22	cin	|- ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1 ..^ ( # ` f ) ) ) )
24		c0	|- (/)
25	23 24	wceq	|- ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1 ..^ ( # ` f ) ) ) ) = (/)
26	10 18 25	w3a	|- ( f ( Trails ` g ) p /\ Fun `' ( p |` ( 1 ..^ ( # ` f ) ) ) /\ ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1 ..^ ( # ` f ) ) ) ) = (/) )
27	26 3 4	copab	|- { <. f , p >. | ( f ( Trails ` g ) p /\ Fun `' ( p |` ( 1 ..^ ( # ` f ) ) ) /\ ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1 ..^ ( # ` f ) ) ) ) = (/) ) }
28	1 2 27	cmpt	|- ( g e. _V |-> { <. f , p >. | ( f ( Trails ` g ) p /\ Fun `' ( p |` ( 1 ..^ ( # ` f ) ) ) /\ ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1 ..^ ( # ` f ) ) ) ) = (/) ) } )
29	0 28	wceq	|- Paths = ( g e. _V |-> { <. f , p >. | ( f ( Trails ` g ) p /\ Fun `' ( p |` ( 1 ..^ ( # ` f ) ) ) /\ ( ( p " { 0 , ( # ` f ) } ) i^i ( p " ( 1 ..^ ( # ` f ) ) ) ) = (/) ) } )