Regular value theorem

Statement

Let $M,N$ be differential manifolds and $p\in N$ be a regular value of a differentiable map $f:M\to N$. Then $f^{-1}(p)$ is a submanifold of $M$.

A slightly stronger version of this result states the following: if there is an open neighbourhood $U$ of $p$ in $N$ such that the rank of the Jacobian is constant for all points in $f^{-1}(U)$, then $f^{-1}(p)$ is a submanifold of $M$.