Theorem spthsfval

Description: The set of simple paths (in an undirected graph). (Contributed by Alexander van der Vekens, 21-Oct-2017) (Revised by AV, 9-Jan-2021) (Revised by AV, 29-Oct-2021)

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

Proof

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