Description: The set ( PathsG ) of all paths on G is a set of pairs by our definition of a path, and so is a relation. (Contributed by AV, 30-Oct-2021)
| Ref | Expression | ||
|---|---|---|---|
| Assertion | relpths | ⊢ Rel ( Paths ‘ 𝐺 ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | df-pths | ⊢ Paths = ( 𝑔 ∈ V ↦ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ Fun ◡ ( 𝑝 ↾ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ∧ ( ( 𝑝 “ { 0 , ( ♯ ‘ 𝑓 ) } ) ∩ ( 𝑝 “ ( 1 ..^ ( ♯ ‘ 𝑓 ) ) ) ) = ∅ ) } ) | |
| 2 | 1 | relmptopab | ⊢ Rel ( Paths ‘ 𝐺 ) |