Metamath Proof Explorer


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 ) )