Metamath Proof Explorer


Theorem spthsfval

Description: The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017) (Revised by AV, 9-Jan-2021) (Revised by AV, 29-Oct-2021)

Ref Expression
Assertion spthsfval ( SPaths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ Fun 𝑝 ) }

Proof

Step Hyp Ref Expression
1 biidd ( ( ⊤ ∧ 𝑔 = 𝐺 ) → ( Fun 𝑝 ↔ Fun 𝑝 ) )
2 wksv { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Walks ‘ 𝐺 ) 𝑝 } ∈ V
3 trliswlk ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝𝑓 ( Walks ‘ 𝐺 ) 𝑝 )
4 3 ssopab2i { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Trails ‘ 𝐺 ) 𝑝 } ⊆ { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Walks ‘ 𝐺 ) 𝑝 }
5 2 4 ssexi { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Trails ‘ 𝐺 ) 𝑝 } ∈ V
6 5 a1i ( ⊤ → { ⟨ 𝑓 , 𝑝 ⟩ ∣ 𝑓 ( Trails ‘ 𝐺 ) 𝑝 } ∈ V )
7 df-spths SPaths = ( 𝑔 ∈ V ↦ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝑔 ) 𝑝 ∧ Fun 𝑝 ) } )
8 1 6 7 fvmptopab ( ⊤ → ( SPaths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ Fun 𝑝 ) } )
9 8 mptru ( SPaths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ Fun 𝑝 ) }