Product of two filters $\mathcal{A}$ and $\mathcal{B}$ is defined as the filter $\mathcal{A}\times\mathcal{B}$ generated by the base $$\{A\times B \,|\, A\in\mathcal{A}, A\in\mathcal{B} \}.$$
I call a filter $\mathcal{F}$ on a set $U$ Cauchy with respect to a fixed filter $\nu$ on the set $U\times U$ iff $\mathcal{F}\times\mathcal{F} \supseteq \nu$.
Let $\mathcal{F}$ be a Cauchy (with respect to $\nu$) proper filter. Prove that there exists a unique minimal filter $x\subseteq\mathcal{F}$ such that $x$ is Cauchy for $\nu$.
Wikipedia says that this statement is true for the special case of $\nu$ being a uniformity.
However, I don't know a proof neither for the general case nor for the special case of $\nu$ being a uniformity. Please help with the proof.
I have proved this theorem for uniform spaces (and some more general settings) in this my preliminary draft article:
http://www.mathematics21.org/binaries/cauchy.pdf
Note that it is a rough draft and I yet need to edit it for clarity and completeness.