Proposition 5.1.8. Let $X$ be a set and $\fB \subset 2^{X \times X}$ be a non-empty family of sets. If
For each $U, V \in \fB$, there exists $W \in \fB$ such that $W \subset U \cap V$.
For each $V \in \fB$, $\Delta \subset V$.
For each $V \in \fB$, there exists $W \in \fB$ such that $W \circ W \subset V$.
then there exists a unique uniformity $\fU \subset 2^{X \times X}$, which is given by
Proof. (F1): By definition of $\fU$.
(F2): For any $U, V \in \fU$, there exists $U_{0}, V_{0} \in \fB$ such that $U_{0} \subset U$ and $V_{0} \subset V$. By (FB1), there exists $W \in \fB$ with $W \subset U_{0} \cap V_{0} \subset U \cap V$. Thus $U \cap V \in \fU$.
(U1) and (U2): For any $U \in \fU$, there exists $U_{0} \in \fB$ with $U_{0} \subset U$. By (UB1), $\Delta \subset U_{0} \subset U$. By (UB2), there exists $V_{0} \in \fB \subset \fU$ with $V_{0} \circ V_{0} \subset U_{0} \subset U$.$\square$