Theorem spthson

Description: The set of simple paths between two vertices (in an undirected graph). (Contributed by Alexander van der Vekens, 1-Mar-2018) (Revised by AV, 16-Jan-2021) (Revised by AV, 21-Mar-2021)

	Ref	Expression
Hypothesis	pthsonfval.v	|- V = ( Vtx ` G )
Assertion	spthson	|- ( ( A e. V /\ B e. V ) -> ( A ( SPathsOn ` G ) B ) = { <. f , p >. | ( f ( A ( TrailsOn ` G ) B ) p /\ f ( SPaths ` G ) p ) } )

Proof

Step	Hyp	Ref	Expression
1		pthsonfval.v	|- V = ( Vtx ` G )
2	1	1vgrex	|- ( A e. V -> G e. _V )
3	2	adantr	|- ( ( A e. V /\ B e. V ) -> G e. _V )
4		simpl	|- ( ( A e. V /\ B e. V ) -> A e. V )
5	4 1	eleqtrdi	|- ( ( A e. V /\ B e. V ) -> A e. ( Vtx ` G ) )
6		simpr	|- ( ( A e. V /\ B e. V ) -> B e. V )
7	6 1	eleqtrdi	|- ( ( A e. V /\ B e. V ) -> B e. ( Vtx ` G ) )
8		wksv	|- { <. f , p >. | f ( Walks ` G ) p } e. _V
9	8	a1i	|- ( ( A e. V /\ B e. V ) -> { <. f , p >. | f ( Walks ` G ) p } e. _V )
10		spthiswlk	|- ( f ( SPaths ` G ) p -> f ( Walks ` G ) p )
11	10	adantl	|- ( ( ( A e. V /\ B e. V ) /\ f ( SPaths ` G ) p ) -> f ( Walks ` G ) p )
12		df-spthson	|- SPathsOn = ( g e. _V |-> ( a e. ( Vtx ` g ) , b e. ( Vtx ` g ) |-> { <. f , p >. | ( f ( a ( TrailsOn ` g ) b ) p /\ f ( SPaths ` g ) p ) } ) )
13	3 5 7 9 11 12	mptmpoopabovd	|- ( ( A e. V /\ B e. V ) -> ( A ( SPathsOn ` G ) B ) = { <. f , p >. | ( f ( A ( TrailsOn ` G ) B ) p /\ f ( SPaths ` G ) p ) } )