Defining a functional map

I am trying to define in latex the following function:

We also want everything to be aligned.

I tried including another align environment but it seems this is not possible:

\begin{align*} \Psi: C([0,1],\mathbb{R}) &\longrightarrow C([0,1],\mathbb{R}) \\ f & \longmapsto \Psi f:\begin{align*}[0,1]&\longrightarrow \mathbb{R}\\x &\longmapsto (\Psi(f))(x)=\int_0^x\!f(y)\,\mathrm{d}y\end{align*}\end{align*}

How to fix this problem?