Order statistics of Uniform$(0,\theta)$ — a $\theta$-free conditional law

Suppose that $X_1,X_2,\dots,X_n$ are iid random variables such that $X_1$ has uniform distribution over $(0,\theta), \theta>0$. Then, which of the following statements is true? (I) The conditional density $f_y(\cdot)$ of $X_{(1)}\mid X_{(n)}=y$ is given by $f_y(x)=(n-1)\frac1y\left(1-\frac xy\right)^{n-2}$ for $0<x<y$, and $0$ otherwise. (II) The conditional distribution of $X_{(1)}\mid X_{(n)}=y$ does not depend on $\theta$.