Local submersion theorem

Statement

Let $M$ and $N$ be differential manifolds of dimensions $m>n$. Let $f:M\to N$ be a differentiable map from $M$ to $N$. Let $p$ be a point such that $f$ is a submersion at $p$, i.e. the induced map on tangent spaces is surjective at $p$. Then, there exists a neighbourhood $U\ni p$ in $M$ and $V\ni f(p)$ in $N$ and coordinate charts for $U$ and $V$ such that in those coordinate charts, the map is a projection map:

$(x_1,x_2,\dots ,x_m)\mapsto (x_1,x_2,\dots ,x_n)$