2spthd

Description: A simple path of length 2 from one vertex to another, different vertex via a third vertex. (Contributed by Alexander van der Vekens, 1-Feb-2018) (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 )
	2spthd.n	\|- ( ph -> A =/= C )
Assertion	2spthd	\|- ( ph -> F ( SPaths ` 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		2spthd.n	\|- ( ph -> A =/= C )
10	1 2 3 4 5 6 7 8	2trld	\|- ( ph -> F ( Trails ` G ) P )
11		3anan32	\|- ( ( A =/= B /\ A =/= C /\ B =/= C ) <-> ( ( A =/= B /\ B =/= C ) /\ A =/= C ) )
12	4 9 11	sylanbrc	\|- ( ph -> ( A =/= B /\ A =/= C /\ B =/= C ) )
13		funcnvs3	\|- ( ( ( A e. V /\ B e. V /\ C e. V ) /\ ( A =/= B /\ A =/= C /\ B =/= C ) ) -> Fun `' <" A B C "> )
14	3 12 13	syl2anc	\|- ( ph -> Fun `' <" A B C "> )
15	1	a1i	\|- ( ph -> P = <" A B C "> )
16	15	cnveqd	\|- ( ph -> `' P = `' <" A B C "> )
17	16	funeqd	\|- ( ph -> ( Fun `' P <-> Fun `' <" A B C "> ) )
18	14 17	mpbird	\|- ( ph -> Fun `' P )
19		isspth	\|- ( F ( SPaths ` G ) P <-> ( F ( Trails ` G ) P /\ Fun `' P ) )
20	10 18 19	sylanbrc	\|- ( ph -> F ( SPaths ` G ) P )

Metamath Proof Explorer

Theorem 2spthd

Proof