1trld

Description: In a graph with two vertices and an edge connecting these two vertices, to go from one vertex to the other vertex via this edge is a trail. The two vertices need not be distinct (in the case of a loop). (Contributed by Alexander van der Vekens, 3-Dec-2017) (Revised by AV, 22-Jan-2021) (Revised by AV, 23-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Hypotheses	1wlkd.p	\|- P = <" X Y ">
	1wlkd.f	\|- F = <" J ">
	1wlkd.x	\|- ( ph -> X e. V )
	1wlkd.y	\|- ( ph -> Y e. V )
	1wlkd.l	\|- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )
	1wlkd.j	\|- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )
	1wlkd.v	\|- V = ( Vtx ` G )
	1wlkd.i	\|- I = ( iEdg ` G )
Assertion	1trld	\|- ( ph -> F ( Trails ` G ) P )

Proof

Step	Hyp	Ref	Expression
1		1wlkd.p	\|- P = <" X Y ">
2		1wlkd.f	\|- F = <" J ">
3		1wlkd.x	\|- ( ph -> X e. V )
4		1wlkd.y	\|- ( ph -> Y e. V )
5		1wlkd.l	\|- ( ( ph /\ X = Y ) -> ( I ` J ) = { X } )
6		1wlkd.j	\|- ( ( ph /\ X =/= Y ) -> { X , Y } C_ ( I ` J ) )
7		1wlkd.v	\|- V = ( Vtx ` G )
8		1wlkd.i	\|- I = ( iEdg ` G )
9	1 2 3 4 5 6 7 8	1wlkd	\|- ( ph -> F ( Walks ` G ) P )
10		funcnvs1	\|- Fun `' <" J ">
11	2	cnveqi	\|- `' F = `' <" J ">
12	11	funeqi	\|- ( Fun `' F <-> Fun `' <" J "> )
13	10 12	mpbir	\|- Fun `' F
14		istrl	\|- ( F ( Trails ` G ) P <-> ( F ( Walks ` G ) P /\ Fun `' F ) )
15	9 13 14	sylanblrc	\|- ( ph -> F ( Trails ` G ) P )

Metamath Proof Explorer

Theorem 1trld

Proof