My lecture notes of classical algebraic geometry on complex field has presented a following result.
Theorem. Let $X$ and $Y$ be (quasi-projective irreducible) varieties, and $f \colon X \to Y$ a morphism between them. Now $f$ could be restricted into smooth morphisms, i.e. there exists $X = \cup_{i} X_i$, $Y = \cup_{j} Y_j$ such that $X_i$ and $Y_i$ are smooth (irreducible) subvarieties of $X$ and $Y$ respectively, and for every $i$, there exists $j = j(i)$ such that $f(X_i) = Y_j$ and the restriction of $f$ on $X_i$ is a smooth morphism.
The proof is somewhat complicated, which first proves that projection has this property by induction on the dimension of a constructible set in $Y$. In the proof, we use Sard's theorem of varieties. My question is:
Could anyone give me some reference about this theorem? I could find it nowhere except my lecture notes. Moreover, I wonder if this problem could be generalized to other cases, for example schemes or varieties over other fields.
Any advice is appreciated.