Theorem trlsfval

Description: The set of trails (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	trlsfval	⊢ ( Trails ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ Fun ◡ 𝑓 ) }

Proof

Step	Hyp	Ref	Expression
1		biidd	⊢ ( 𝑔 = 𝐺 → ( Fun ◡ 𝑓 ↔ Fun ◡ 𝑓 ) )
2		df-trls	⊢ Trails = ( 𝑔 ∈ V ↦ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Walks ‘ 𝑔 ) 𝑝 ∧ Fun ◡ 𝑓 ) } )
3	1 2	fvmptopab	⊢ ( Trails ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ Fun ◡ 𝑓 ) }