Description: A cycle is a circuit. (Contributed by Alexander van der Vekens, 30-Oct-2017) (Revised by AV, 31-Jan-2021) (Proof shortened by AV, 30-Oct-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | cycliscrct | |- ( F ( Cycles ` G ) P -> F ( Circuits ` G ) P ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pthistrl | |- ( F ( Paths ` G ) P -> F ( Trails ` G ) P ) |
|
| 2 | 1 | anim1i | |- ( ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) -> ( F ( Trails ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) |
| 3 | iscycl | |- ( F ( Cycles ` G ) P <-> ( F ( Paths ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) |
|
| 4 | iscrct | |- ( F ( Circuits ` G ) P <-> ( F ( Trails ` G ) P /\ ( P ` 0 ) = ( P ` ( # ` F ) ) ) ) |
|
| 5 | 2 3 4 | 3imtr4i | |- ( F ( Cycles ` G ) P -> F ( Circuits ` G ) P ) |