If $X$ and $Y$ are independent random variables, then $f_{X|Y}(x|y) = f_X(x)$. Is the converse false?
I think it's false because suppose you have a set of i.i.d. random variables, $\{X_i\}$, with common CDF $F$. Then the CDF given the minimum, $X_{(1)}$, is $F(x) - F(X_1)$ if $x \geq X_1$. If the $X_i$s have density, then $f_{X_1|X_{(1)}}(x) = f(x)$ because $F(X_1)$ is constant with respect to $x$. But $X_1$ and $X_{(1)}$ are not independent, right?