Let $X,Y$ be complete metric spaces, and let $\Sigma:X\times Y\rightarrow Y$ be a continous mapping which satisfies the following property: there exists a $C<1$, such that for all $x\in X$ and $y_{1},y_{2}\in Y$ one has $d(\Sigma(x,y_{1}),\Sigma(x,y_{2}))\leq Cd(y_{1},y_{2})$. The fixed point theorem for complete metric spaces then implies that for all $x\in X$, there exists a unique $y(x)\in Y$ such that $\Sigma(x,y(x))=y(x)$. It is possible to prove that the correspondence $x\mapsto y(x)$ is a continuous map of metric spaces $X\rightarrow Y$.

So far I stated what I know about general metric spaces. Now furtherly assume that in particular, $X,Y$ are Fréchet spaces, and that $\Sigma$ is a smooth map of Fréchet manifolds.

My question is this: can we conclude under these additional assumptions that the correspondence $x\mapsto y(x)$ is actually a smooth map between Fréchet manifolds? I haven't found any reference for this, nor could I find a way to extend the continuous-metric proof to the smooth setting.

I will be extremely thankful for guidance or reference.