A submanifold (of $\mathbb{R}^n$), it appears, can be defined in several equivalent ways. One definition, paraphrased from Amann and Escher's Analysis II, is as follows:
A subset $M$ of $\mathbb{R}^n$ is said to be a smooth $m$-dimensional submanifold of $\mathbb{R}^n$ if for every $x \in M$ there exists an open neighborhood $U$ of $x$, an open set $V$ in $\mathbb{R}^n$ and a smooth diffeomorphism $\phi:U \rightarrow V$ such that $\phi(U \cap M) = V \cap (\mathbb{R}^m \times \{0\})$
Everything about this definition makes sense to me except the intersection of $V$ with the Cartesian product of $\mathbb{R}^m$ and $\{0\}$ instead of just $\mathbb{R}^m$. Why not just require $\phi(U \cap M) = V \cap (\mathbb{R}^m)$?