Uniformity is generated by pseudometrics

Question

How to prove that every uniform space is generated by a family of pseudometric spaces?

You may offer me a book. In Engelking this theorem is presented without a proof. In Willard it is a exercise. Can you propose me a book really having a proof of this theorem? Direct proof at MSE would be even better.

Answer 1

From Bourbaki's General topology. Chapters 5 to 10, theorem 1 page 142:

Theorem: Given an uniformity $\mathcal{U}$ on a set $X$, there is a family of pseudometrics on $X$ such that the uniformity defined by this family is identical with $\mathcal{U}$.