Preimage theorem

In mathematics, particularly in differential topology, the preimage theorem is a theorem concerning the preimage of particular points in a manifold under the action of a smooth map.

Statement of Theorem
Definition. Let $$f: X \to Y\,\!$$ be a smooth map between manifolds. We say that a point $$y \in Y$$ is a regular value of f if for all $$x \in f^{-1}(y)$$ the map $$df_x: T_xX \to T_yY\,\!$$ is surjective. Here, $$T_xX\,\!$$ and $$T_yY\,\!$$ are the tangent spaces of X and Y at the points x and y.

Theorem. Let $$f: X \to Y\,\!$$ be a smooth map, and let $$y \in Y$$ be a regular value of f. Then $$\{x : x \in f^{-1}(y)\}$$ is a submanifold of X. Further, the codimension of this manifold in X is equal to the dimension of Y.