Theorem trlsfval

Description: The set of trails (in an undirected graph). (Contributed by Alexander van der Vekens, 20-Oct-2017) (Revised by AV, 28-Dec-2020) (Revised by AV, 29-Oct-2021)

	Ref	Expression
Assertion	trlsfval	$$⊢\mathrm{Trails}\left(G\right)=\left\{⟨f,p⟩|\left(f\mathrm{Walks}\left(G\right)p\wedge \mathrm{Fun}{f}^{-1}\right)\right\}$$

Proof

Step	Hyp	Ref	Expression
1		biidd	$$⊢g=G\to \left(\mathrm{Fun}{f}^{-1}\leftrightarrow \mathrm{Fun}{f}^{-1}\right)$$
2		df-trls	$$⊢\mathrm{Trails}=\left(g\in \mathrm{V}⟼\left\{⟨f,p⟩|\left(f\mathrm{Walks}\left(g\right)p\wedge \mathrm{Fun}{f}^{-1}\right)\right\}\right)$$
3	1 2	fvmptopab	$$⊢\mathrm{Trails}\left(G\right)=\left\{⟨f,p⟩|\left(f\mathrm{Walks}\left(G\right)p\wedge \mathrm{Fun}{f}^{-1}\right)\right\}$$