Theorem istrl

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 ) )

Proof

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 ) )