Theorem relpths

Description: The set ( PathsG ) of all paths on G is a set of pairs by our definition of a path, and so is a relation. (Contributed by AV, 30-Oct-2021)

	Ref	Expression
Assertion	relpths	$$⊢\mathrm{Rel}\mathrm{Paths}\left(G\right)$$

Proof

Step	Hyp	Ref	Expression
1		df-pths	$$⊢\mathrm{Paths}=\left(g\in \mathrm{V}⟼\left\{⟨f,p⟩|\left(f\mathrm{Trails}\left(g\right)p\wedge \mathrm{Fun}{\left({p\upharpoonright }_{\left(1..^\left|f\right|\right)}\right)}^{-1}\wedge p\left[\left\{0,\left|f\right|\right\}\right]\cap p\left[\left(1..^\left|f\right|\right)\right]=\varnothing \right)\right\}\right)$$
2	1	relmptopab	$$⊢\mathrm{Rel}\mathrm{Paths}\left(G\right)$$