# Why does the cocycle condition on fiber isomorphisms imply a fiber bundle is trivial?

While reading about connections I stumbled on this entry of the encyclopedia of math. In the second paragraph of the comments section it's written that if a fiber bundle admits coherent isomorphisms of its fibers, where "coherent" means "satisfying the cocycle condition", then the fiber bundle is actually trivial.

What's the geometric intuition behind this fact, and how is it proved?

To prove it suffice to take a neighborhood over a fixed point $b$,using Zorn's lemma to find a largest open set $U$ containing $b$ over which the bundle is trivial. Then pick up a point on the boundary of $U$ and a small neighborhood around it. By the cocycle condition all the fibers above points over $U$ which has been identified vertically over the base can be "moved" to the new small neighborhood. Thus if we incorporate the small neighbhorhood into $U$ to enlarge the fiber bundle using the isomorphism of fibers, we face no difficulty. This contradicts the maximality of $U$. So $U=B$ and $E$ is isomorphic to the product bundle as desired.