A question about the proof of the regular value theorem

Question

I saw the claim, to prove the regular value theorem, It is enough to show that for every point a of $f^{-1}$(c) there is a neighborhood U of a such that U ∩ $f^{−1}$(c) is a submanifold of U of dimension m − n (where c is the regular point). I wonder how to show this guarantees that $f^{-1}$(c) is a manifold?