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Database GRAPH THEORY Walks, paths and cycles Closed walks 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 ) )