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Metamath Proof Explorer

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 ) }