I'm having trouble understanding the first condition of the Jacobian criterion for smoothness:
Theorem: (Jacobian criterion)
Given the following commutative diagram of schemes,
$\begin{array}{ccccccccc} Z & \xrightarrow{i} & \mathbb{A}_R^n \\\ &{f}\searrow & \downarrow{j} \\\ && \operatorname{Spec}R \end{array} $
where $i$ is a closed immersion which is locally of finite presentation, then for any $z\in Z$, $f$ is smooth at $z$ if and only if $Z=V(f_1,\ldots,f_r)$ locally around $z$, with $f_1,\ldots,f_r\in R[X_1,\ldots,X_r]$, and the rank of the Jacobi matrix $\mathcal{J}(z)=\left(\frac{\partial f_i(z)}{\partial X_j}\right)$ is $r$.
Why is $Z=V(f_1,\ldots,f_r)$ stated explicitly as a condition? Doesn't this already follow from the fact that $i$ is a closed immersion?