Theorem 2trld

Description: Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 4-Dec-2017) (Revised by AV, 24-Jan-2021) (Revised by AV, 24-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Hypotheses	2wlkd.p	|- P = <" A B C ">
	2wlkd.f	|- F = <" J K ">
	2wlkd.s	|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
	2wlkd.n	|- ( ph -> ( A =/= B /\ B =/= C ) )
	2wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
	2wlkd.v	|- V = ( Vtx ` G )
	2wlkd.i	|- I = ( iEdg ` G )
	2trld.n	|- ( ph -> J =/= K )
Assertion	2trld	|- ( ph -> F ( Trails ` G ) P )

Proof

Step	Hyp	Ref	Expression
1		2wlkd.p	|- P = <" A B C ">
2		2wlkd.f	|- F = <" J K ">
3		2wlkd.s	|- ( ph -> ( A e. V /\ B e. V /\ C e. V ) )
4		2wlkd.n	|- ( ph -> ( A =/= B /\ B =/= C ) )
5		2wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) ) )
6		2wlkd.v	|- V = ( Vtx ` G )
7		2wlkd.i	|- I = ( iEdg ` G )
8		2trld.n	|- ( ph -> J =/= K )
9	1 2 3 4 5 6 7	2wlkd	|- ( ph -> F ( Walks ` G ) P )
10	1 2 3 4 5	2wlkdlem7	|- ( ph -> ( J e. _V /\ K e. _V ) )
11		df-3an	|- ( ( J e. _V /\ K e. _V /\ J =/= K ) <-> ( ( J e. _V /\ K e. _V ) /\ J =/= K ) )
12	10 8 11	sylanbrc	|- ( ph -> ( J e. _V /\ K e. _V /\ J =/= K ) )
13		funcnvs2	|- ( ( J e. _V /\ K e. _V /\ J =/= K ) -> Fun `' <" J K "> )
14	12 13	syl	|- ( ph -> Fun `' <" J K "> )
15	2	cnveqi	|- `' F = `' <" J K ">
16	15	funeqi	|- ( Fun `' F <-> Fun `' <" J K "> )
17	14 16	sylibr	|- ( ph -> Fun `' F )
18		istrl	|- ( F ( Trails ` G ) P <-> ( F ( Walks ` G ) P /\ Fun `' F ) )
19	9 17 18	sylanbrc	|- ( ph -> F ( Trails ` G ) P )