Metamath Proof Explorer

Theorem wlkv

Description: The classes involved in a walk are sets. (Contributed by Alexander van der Vekens, 31-Oct-2017) (Revised by AV, 3-Feb-2021)

Ref Expression
Assertion wlkv 
|- ( F ( Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )

Proof

Step Hyp Ref Expression
1 eqid 
 |-  ( Vtx ` G ) = ( Vtx ` G )
2 eqid 
 |-  ( iEdg ` G ) = ( iEdg ` G )
3 1 2 wksfval 
 |-  ( G e. _V -> ( Walks ` G ) = { <. f , p >. | ( f e. Word dom ( iEdg ` G ) /\ p : ( 0 ... ( # ` f ) ) --> ( Vtx ` G ) /\ A. k e. ( 0 ..^ ( # ` f ) ) if- ( ( p ` k ) = ( p ` ( k + 1 ) ) , ( ( iEdg ` G ) ` ( f ` k ) ) = { ( p ` k ) } , { ( p ` k ) , ( p ` ( k + 1 ) ) } C_ ( ( iEdg ` G ) ` ( f ` k ) ) ) ) } )
4 3 brfvopab 
 |-  ( F ( Walks ` G ) P -> ( G e. _V /\ F e. _V /\ P e. _V ) )