Propierties of the submersion

Question

Let $p:M\rightarrow N$ a surjective submersion of manifolds. Let $Y\subset N$ I want to prove that

$Y$ is a submanifold of $N$ if and only if $p^{-1}(Y)$ is a submanifold $M$.

I could prove that if $Y$ is a submanifold then $p^{-1}(Y)$ using the regular point theorem, but I could not prove the converse.

Any hint?

Answer 1

You can directly show that something is a manifold by exhibiting a system of charts for it with the right transition maps. Can you find a way of taking the system of charts on $p^{-1}(Y)$ and moving them over to give a system of charts on $Y$?