I know the definition of a fiber bundle as a map $p:E \rightarrow B$ such that the preimage of any open set in $B$ is diffeomorphic to a product and fits in a commutative diagram with the projection $p$.
If we have a manifold $E$ and we know that it is locally a product, i.e. for every point $p \in E$ there exist an open set $V$ containing $p$ such that $V \simeq U \times F$ where $F$ is some fixed manifold and $U$ is an open set of $\mathbb{R}^n$ for a fixed $n$. Is that enough to see that $E$ is a fiber bundle in the above sense? In that case, how do we construct $B$? If not, could anyone provide a counterexample?
Also it is considered a good answer if someone provides a good reference where this is explained.