We have $f: X \to Y$ an etale morphism of varieties (separated, irreducible, reduced scheme of finite type over $k$) over $k$, an algebraically closed field. Why is $X$ locally the locus of roots of $n$ equations in $Y \times \mathbb{A}^n$ given etale $f: X \to Y$?
Given such $f$ we know: there exist affine opens $\operatorname{Spec}{A} = U \subseteq X$ and $\operatorname{Spec}{R} = V \subseteq Y$ with $x \in U$, corresponding to prime ideal $Q$, and $f(U) \subseteq V$ such that there exists a presentation $A = R[X_1, \ldots, X_n]/(f_1, \ldots, f_n)$ with $$ \det (\partial f_i / \partial X_j) + (f_1, \ldots, f_n) \notin Q. $$
How do I see form this that $U$ is cut out by $n$ equations in $Y \times \mathbb{A}^n$? Thank you.