If I have $n$ independent, identically distributed uniform $(a,b)$ random variables, why is this true: $$ \max(x\_i) \mid \min(x\_i) \sim \mathrm{Uniform}(\min(x\_i),b) $$

I agree that the probability density function of $\max(x\_i) \mid \min(x\_i)$ must be non-zero only in the range $[\min(x\_i), b]$.

I also find it reasonable that the posterior distribution is uniform, but I cannot see why it has to be so.

Also, I am confused how to derive this algebraically. Let $X$ be the max of $n$ iid uniform random variables. Let $Y$ be the min of $n$ iid uniform random variables. If I do $$ f\_{X|Y}(x|y) = \frac{f\_{X,Y}(x,y)}{f\_Y(y)}, $$
$f\_Y(y)$ is known. But the joint distribution is unknown ($X$ and $Y$ are not independent).

Thanks.