Prove the following theorem.
Fix $r\in \{0\}\cup\{\mathbb{N}\}\cup\{\infty\}$. A subset $M \subseteq \mathbb{R}^k$ is a $C^r$-manifolf (without boundry) of dimension $m$ if and only if for all points $c \in M$ there exists
a. an open neighborhood $V \subseteq \mathbb{R}^k$ of $c$
b. an open set $U \subseteq \mathbb{R}^m$
c. $C^r$ diffeomorphism $U \xleftarrow {\xrightarrow{\phi}}_{\psi} V\cap M$
My attempt:
By definition we know that subset $M \subseteq \mathbb{R}^k$ is a $C^r$-manifolf (without boundry) of dimension $m$ if and only if for all points $c \in M$ there exists an open neighborhood $U \subset \mathbb{R}^k$, an open set $V \subseteq \mathbb{R}^k$ with a $C^r$ diffeomorphism $\phi:U \rightarrow V$ such that $\phi(U \cap M) = V \cap (\mathbb{R}^m \times \{0\})$ with inverse $\psi:V \rightarrow U$.
Not sure where to go from here.