Preimage Theorem

Theorem
Let $y$ be a regular value of a smooth submersion $f:X \to Y$.

Then the preimage $f^{-1}(y)$ is a smooth submanifold of $X$, with $\dim f^{-1}(y) = \dim X - \dim Y$.

Proof
Let $k,l$ be natural numbers with $k \geq l$.

By the Local Submersion Theorem, there exists coordinates in some open sets of $x,y$ such that $f(x_1, x_2, \ldots, x_k)=(x_1, \ldots,x_l)$ and $y$ corresponds to $(0, \ldots, 0)$.

Let $V$ be that neighborhood of $x$.

Then $f^{-1}(y) \cap V$ is the set of points where $x_1=0, \ldots, x_l=0$.

The functions $x_{l+1}, \ldots, x_k$ therefore form a coordinate system on the set $f^{-1}(y) \cap V$, which is a relatively open subset of $f^{-1}(y)$.

Together these functions then form a diffeomorphism to a Euclidean space.

We also have, by the regular value properties of $y$, a surjection of tangent spaces from $x$ to $y$.

This ensures smoothness of the solution set $f^{-1}(y)$.