Theorem wlkvtxedg

Description: The vertices of a walk are connected by edges. (Contributed by Alexander van der Vekens, 22-Jul-2018) (Revised by AV, 2-Jan-2021)

	Ref	Expression
Hypothesis	wlkvtxedg.e	|- E = ( Edg ` G )
Assertion	wlkvtxedg	|- ( F ( Walks ` G ) P -> A. k e. ( 0 ..^ ( # ` F ) ) E. e e. E { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )

Proof

Step	Hyp	Ref	Expression
1		wlkvtxedg.e	|- E = ( Edg ` G )
2		eqid	|- ( iEdg ` G ) = ( iEdg ` G )
3	2	wlkvtxiedg	|- ( F ( Walks ` G ) P -> A. k e. ( 0 ..^ ( # ` F ) ) E. e e. ran ( iEdg ` G ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )
4		edgval	|- ( Edg ` G ) = ran ( iEdg ` G )
5	1 4	eqtr2i	|- ran ( iEdg ` G ) = E
6	5	rexeqi	|- ( E. e e. ran ( iEdg ` G ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e <-> E. e e. E { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )
7	6	ralbii	|- ( A. k e. ( 0 ..^ ( # ` F ) ) E. e e. ran ( iEdg ` G ) { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e <-> A. k e. ( 0 ..^ ( # ` F ) ) E. e e. E { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )
8	3 7	sylib	|- ( F ( Walks ` G ) P -> A. k e. ( 0 ..^ ( # ` F ) ) E. e e. E { ( P ` k ) , ( P ` ( k + 1 ) ) } C_ e )