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	|- ( ( T. /\ g = G ) -> ( Fun `' f <-> Fun `' f ) )
2		wksv	|- { <. f , p >. | f ( Walks ` G ) p } e. _V
3	2	a1i	|- ( T. -> { <. f , p >. | f ( Walks ` G ) p } e. _V )
4		df-trls	|- Trails = ( g e. _V |-> { <. f , p >. | ( f ( Walks ` g ) p /\ Fun `' f ) } )
5	1 3 4	fvmptopab	|- ( T. -> ( Trails ` G ) = { <. f , p >. | ( f ( Walks ` G ) p /\ Fun `' f ) } )
6	5	mptru	|- ( Trails ` G ) = { <. f , p >. | ( f ( Walks ` G ) p /\ Fun `' f ) }