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	|- ( ( T. /\ g = G ) -> ( Fun `' p <-> Fun `' p ) )
2		wksv	|- { <. f , p >. | f ( Walks ` G ) p } e. _V
3		trliswlk	|- ( f ( Trails ` G ) p -> f ( Walks ` G ) p )
4	3	ssopab2i	|- { <. f , p >. | f ( Trails ` G ) p } C_ { <. f , p >. | f ( Walks ` G ) p }
5	2 4	ssexi	|- { <. f , p >. | f ( Trails ` G ) p } e. _V
6	5	a1i	|- ( T. -> { <. f , p >. | f ( Trails ` G ) p } e. _V )
7		df-spths	|- SPaths = ( g e. _V |-> { <. f , p >. | ( f ( Trails ` g ) p /\ Fun `' p ) } )
8	1 6 7	fvmptopab	|- ( T. -> ( SPaths ` G ) = { <. f , p >. | ( f ( Trails ` G ) p /\ Fun `' p ) } )
9	8	mptru	|- ( SPaths ` G ) = { <. f , p >. | ( f ( Trails ` G ) p /\ Fun `' p ) }