Theorem 3trld

Description: Construction of a trail from two given edges in a graph. (Contributed by Alexander van der Vekens, 13-Nov-2017) (Revised by AV, 8-Feb-2021) (Revised by AV, 24-Mar-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Hypotheses	3wlkd.p	|- P = <" A B C D ">
	3wlkd.f	|- F = <" J K L ">
	3wlkd.s	|- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
	3wlkd.n	|- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
	3wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )
	3wlkd.v	|- V = ( Vtx ` G )
	3wlkd.i	|- I = ( iEdg ` G )
	3trld.n	|- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )
Assertion	3trld	|- ( ph -> F ( Trails ` G ) P )

Proof

Step	Hyp	Ref	Expression
1		3wlkd.p	|- P = <" A B C D ">
2		3wlkd.f	|- F = <" J K L ">
3		3wlkd.s	|- ( ph -> ( ( A e. V /\ B e. V ) /\ ( C e. V /\ D e. V ) ) )
4		3wlkd.n	|- ( ph -> ( ( A =/= B /\ A =/= C ) /\ ( B =/= C /\ B =/= D ) /\ C =/= D ) )
5		3wlkd.e	|- ( ph -> ( { A , B } C_ ( I ` J ) /\ { B , C } C_ ( I ` K ) /\ { C , D } C_ ( I ` L ) ) )
6		3wlkd.v	|- V = ( Vtx ` G )
7		3wlkd.i	|- I = ( iEdg ` G )
8		3trld.n	|- ( ph -> ( J =/= K /\ J =/= L /\ K =/= L ) )
9	1 2 3 4 5 6 7	3wlkd	|- ( ph -> F ( Walks ` G ) P )
10	1 2 3 4 5	3wlkdlem7	|- ( ph -> ( J e. _V /\ K e. _V /\ L e. _V ) )
11		funcnvs3	|- ( ( ( J e. _V /\ K e. _V /\ L e. _V ) /\ ( J =/= K /\ J =/= L /\ K =/= L ) ) -> Fun `' <" J K L "> )
12	10 8 11	syl2anc	|- ( ph -> Fun `' <" J K L "> )
13	2	cnveqi	|- `' F = `' <" J K L ">
14	13	funeqi	|- ( Fun `' F <-> Fun `' <" J K L "> )
15	12 14	sylibr	|- ( ph -> Fun `' F )
16		istrl	|- ( F ( Trails ` G ) P <-> ( F ( Walks ` G ) P /\ Fun `' F ) )
17	9 15 16	sylanbrc	|- ( ph -> F ( Trails ` G ) P )