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Metamath Proof Explorer

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 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 ) } ) )
9 3 5 7 8 mptmpoopabovd 
 |-  ( ( A e. V /\ B e. V ) -> ( A ( SPathsOn ` G ) B ) = { <. f , p >. | ( f ( A ( TrailsOn ` G ) B ) p /\ f ( SPaths ` G ) p ) } )