Theorem clwlkwlk

Description: Closed walks are walks (in an undirected graph). (Contributed by Alexander van der Vekens, 23-Jun-2018) (Revised by AV, 16-Feb-2021) (Proof shortened by AV, 30-Oct-2021)

	Ref	Expression
Assertion	clwlkwlk	⊢ ( 𝑊 ∈ ( ClWalks ‘ 𝐺 ) → 𝑊 ∈ ( Walks ‘ 𝐺 ) )

Proof

Step	Hyp	Ref	Expression
1		elopabran	⊢ ( 𝑊 ∈ { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ) } → 𝑊 ∈ ( Walks ‘ 𝐺 ) )
2		clwlks	⊢ ( ClWalks ‘ 𝐺 ) = { ⟨ 𝑓 , 𝑝 ⟩ ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ) }
3	1 2	eleq2s	⊢ ( 𝑊 ∈ ( ClWalks ‘ 𝐺 ) → 𝑊 ∈ ( Walks ‘ 𝐺 ) )