Theorem trlsfval

Description: The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017) (Revised by AV, 28-Dec-2020) (Revised by AV, 29-Oct-2021)

	Ref	Expression
Assertion	trlsfval	|- ( Trails ` G ) = { <. f , p >. | ( f ( Walks ` G ) p /\ Fun `' f ) }

Proof

Step	Hyp	Ref	Expression
1		biidd	|- ( g = G -> ( Fun `' f <-> Fun `' f ) )
2		df-trls	|- Trails = ( g e. _V |-> { <. f , p >. | ( f ( Walks ` g ) p /\ Fun `' f ) } )
3	1 2	fvmptopab	|- ( Trails ` G ) = { <. f , p >. | ( f ( Walks ` G ) p /\ Fun `' f ) }