Metamath Proof Explorer

Theorem relwlk

Description: The set ( WalksG ) of all walks on G is a set of pairs by our definition of a walk, and so is a relation. (Contributed by Alexander van der Vekens, 30-Jun-2018) (Revised by AV, 19-Feb-2021)

		Ref	Expression
	Assertion	relwlk	\|- Rel ( Walks ` G )

Proof

Step	Hyp	Ref	Expression
1		df-wlks	\|- Walks = ( g e. _V \|-> { <. f , p >. \| ( f e. Word dom ( iEdg ` g ) /\ p : ( 0 ... ( # ` f ) ) --> ( Vtx ` g ) /\ A. k e. ( 0 ..^ ( # ` f ) ) if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( ( iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( ( iEdg ` g ) ` ( f ` k ) ) ) ) } )
2	1	relmptopab	\|- Rel ( Walks ` G )

Step

Hyp

Ref

Expression

df-wlks

 |-  Walks = ( g e. _V |-> { <. f , p >. | ( f e. Word dom ( iEdg ` g ) /\ p : ( 0 ... ( # ` f ) ) --> ( Vtx ` g ) /\ A. k e. ( 0 ..^ ( # ` f ) ) if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( ( iEdg ` g ) ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( ( iEdg ` g ) ` ( f ` k ) ) ) ) } )

relmptopab

 |-  Rel ( Walks ` G )