Theorem isspth

Description: Conditions for a pair of classes/functions to be a simple path (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	isspth	⊢ ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝑃 ) )

Proof

Step	Hyp	Ref	Expression
1		spthsfval	⊢ ( SPaths ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Trails ‘ 𝐺 ) 𝑝 ∧ Fun ◡ 𝑝 ) }
2		cnveq	⊢ ( 𝑝 = 𝑃 → ◡ 𝑝 = ◡ 𝑃 )
3	2	funeqd	⊢ ( 𝑝 = 𝑃 → ( Fun ◡ 𝑝 ↔ Fun ◡ 𝑃 ) )
4	3	adantl	⊢ ( ( 𝑓 = 𝐹 ∧ 𝑝 = 𝑃 ) → ( Fun ◡ 𝑝 ↔ Fun ◡ 𝑃 ) )
5		reltrls	⊢ Rel ( Trails ‘ 𝐺 )
6	1 4 5	brfvopabrbr	⊢ ( 𝐹 ( SPaths ‘ 𝐺 ) 𝑃 ↔ ( 𝐹 ( Trails ‘ 𝐺 ) 𝑃 ∧ Fun ◡ 𝑃 ) )