I'm using the uniform cover definition of a uniform space.
Let $f:(X, \mu)\to (Y, \nu)$ be a surjective map such that $(X, \mu)$ is a uniform space and $\nu$ is a largest preuniformity on $Y$ such that if $\mathcal{U}\in\nu$ then $f^{-1}(\mathcal{U})\in\mu$.
I want to prove that then $\nu$ is a uniformity. I've proved that if $\mathcal{V}\in \mu$ then $f(\mathcal{V})\in \nu$ (this is a cover since $f$ is surjective).
I've actually proved this hoping that a cover which separates some of the fibers in $X$ will arise and I'd be able to use it to separate two points in $Y$. I'm stuck.