Theorem clwlks

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	$$⊢\mathrm{ClWalks}\left(G\right)=\left\{⟨f,p⟩|\left(f\mathrm{Walks}\left(G\right)p\wedge p\left(0\right)=p\left(\left|f\right|\right)\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		biidd	$$⊢\left(\top \wedge g=G\right)\to \left(p\left(0\right)=p\left(\left|f\right|\right)\leftrightarrow p\left(0\right)=p\left(\left|f\right|\right)\right)$$
2		wksv	$$⊢\left\{⟨f,p⟩|f\mathrm{Walks}\left(G\right)p\right\}\in \mathrm{V}$$
3	2	a1i	$$⊢\top \to \left\{⟨f,p⟩|f\mathrm{Walks}\left(G\right)p\right\}\in \mathrm{V}$$
4		df-clwlks	$$⊢\mathrm{ClWalks}=\left(g\in \mathrm{V}⟼\left\{⟨f,p⟩|\left(f\mathrm{Walks}\left(g\right)p\wedge p\left(0\right)=p\left(\left|f\right|\right)\right)\right\}\right)$$
5	1 3 4	fvmptopab	$$⊢\top \to \mathrm{ClWalks}\left(G\right)=\left\{⟨f,p⟩|\left(f\mathrm{Walks}\left(G\right)p\wedge p\left(0\right)=p\left(\left|f\right|\right)\right)\right\}$$
6	5	mptru	$$⊢\mathrm{ClWalks}\left(G\right)=\left\{⟨f,p⟩|\left(f\mathrm{Walks}\left(G\right)p\wedge p\left(0\right)=p\left(\left|f\right|\right)\right)\right\}$$