Description: Conditions for a pair of classes/functions to be a trail (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017) (Revised by AV, 28-Dec-2020) (Revised by AV, 29-Oct-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | istrl | |- ( F ( Trails ` G ) P <-> ( F ( Walks ` G ) P /\ Fun `' F ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | trlsfval | |- ( Trails ` G ) = { <. f , p >. | ( f ( Walks ` G ) p /\ Fun `' f ) } |
|
| 2 | cnveq | |- ( f = F -> `' f = `' F ) |
|
| 3 | 2 | funeqd | |- ( f = F -> ( Fun `' f <-> Fun `' F ) ) |
| 4 | 3 | adantr | |- ( ( f = F /\ p = P ) -> ( Fun `' f <-> Fun `' F ) ) |
| 5 | relwlk | |- Rel ( Walks ` G ) |
|
| 6 | 1 4 5 | brfvopabrbr | |- ( F ( Trails ` G ) P <-> ( F ( Walks ` G ) P /\ Fun `' F ) ) |