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	$$⊢\left(\top \wedge g=G\right)\to \left(\mathrm{Fun}{p}^{-1}\leftrightarrow \mathrm{Fun}{p}^{-1}\right)$$
2		wksv	$$⊢\left\{⟨f,p⟩|f\mathrm{Walks}\left(G\right)p\right\}\in \mathrm{V}$$
3		trliswlk	$$⊢f\mathrm{Trails}\left(G\right)p\to f\mathrm{Walks}\left(G\right)p$$
4	3	ssopab2i	$$⊢\left\{⟨f,p⟩|f\mathrm{Trails}\left(G\right)p\right\}\subseteq \left\{⟨f,p⟩|f\mathrm{Walks}\left(G\right)p\right\}$$
5	2 4	ssexi	$$⊢\left\{⟨f,p⟩|f\mathrm{Trails}\left(G\right)p\right\}\in \mathrm{V}$$
6	5	a1i	$$⊢\top \to \left\{⟨f,p⟩|f\mathrm{Trails}\left(G\right)p\right\}\in \mathrm{V}$$
7		df-spths	$$⊢\mathrm{SPaths}=\left(g\in \mathrm{V}⟼\left\{⟨f,p⟩|\left(f\mathrm{Trails}\left(g\right)p\wedge \mathrm{Fun}{p}^{-1}\right)\right\}\right)$$
8	1 6 7	fvmptopab	$$⊢\top \to \mathrm{SPaths}\left(G\right)=\left\{⟨f,p⟩|\left(f\mathrm{Trails}\left(G\right)p\wedge \mathrm{Fun}{p}^{-1}\right)\right\}$$
9	8	mptru	$$⊢\mathrm{SPaths}\left(G\right)=\left\{⟨f,p⟩|\left(f\mathrm{Trails}\left(G\right)p\wedge \mathrm{Fun}{p}^{-1}\right)\right\}$$