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