Theorem spthsfval

Description: The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017) (Revised by AV, 9-Jan-2021) (Revised by AV, 29-Oct-2021)

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

Proof

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