Proximal operator definition is: \begin{align} \operatorname{prox}_{\eta f}(x) := \arg\min_{z} \ \eta f(z) + \frac{1}{2}\| z - x \|_2^2, \end{align} where typically $f$ is assumed to be closed convex proper function.
To this end, let $f$ be an $L$-smooth function, i.e., $\|\nabla f(x) - \nabla f(y) \| \leq L \|x-y\|, \ \forall x,y \in \operatorname{dom}(f)$. Then, in this paper (page 4), it states that if $0 < \eta < \frac{1}{L}$ the proximal operator of such a function is well-defined and single-valued.
My question is about this "well-defined and single-valued". Can we say the following is valid for such a smooth nonconvex function as well? \begin{align} p = \operatorname{prox}_{\eta f}(x) \Longleftrightarrow x - p = \eta \nabla f(x). \end{align}
If not, then can someone please enlighten me what does "well-defined and single-valued" mean then?