Database
GRAPH THEORY
Walks, paths and cycles
Closed walks
clwlks
Metamath Proof Explorer
Description: The set of closed walks (in an undirected graph). (Contributed by Alexander van der Vekens , 15-Mar-2018) (Revised by AV , 16-Feb-2021)
(Revised by AV , 29-Oct-2021)
Ref
Expression
Assertion
clwlks
⊢ ( ClWalks ‘ 𝐺 ) = { 〈 𝑓 , 𝑝 〉 ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ) }
Proof
Step
Hyp
Ref
Expression
1
biidd
⊢ ( 𝑔 = 𝐺 → ( ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ↔ ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ) )
2
df-clwlks
⊢ ClWalks = ( 𝑔 ∈ V ↦ { 〈 𝑓 , 𝑝 〉 ∣ ( 𝑓 ( Walks ‘ 𝑔 ) 𝑝 ∧ ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ) } )
3
1 2
fvmptopab
⊢ ( ClWalks ‘ 𝐺 ) = { 〈 𝑓 , 𝑝 〉 ∣ ( 𝑓 ( Walks ‘ 𝐺 ) 𝑝 ∧ ( 𝑝 ‘ 0 ) = ( 𝑝 ‘ ( ♯ ‘ 𝑓 ) ) ) }