Description: The set of circuits (in an undirected graph). (Contributed by Alexander van der Vekens, 30-Oct-2017) (Revised by AV, 31-Jan-2021)
Ref | Expression | ||
---|---|---|---|
Assertion | crcts | |- ( Circuits ` G ) = { <. f , p >. | ( f ( Trails ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | biidd | |- ( g = G -> ( ( p ` 0 ) = ( p ` ( # ` f ) ) <-> ( p ` 0 ) = ( p ` ( # ` f ) ) ) ) |
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2 | df-crcts | |- Circuits = ( g e. _V |-> { <. f , p >. | ( f ( Trails ` g ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } ) |
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3 | 1 2 | fvmptopab | |- ( Circuits ` G ) = { <. f , p >. | ( f ( Trails ` G ) p /\ ( p ` 0 ) = ( p ` ( # ` f ) ) ) } |