Let $p : E \to B$ be a covering map. And also assume that $B$ is path-connected and locally path-connected. Then, given y in $B$, there exists a neighborhood $U$ of $y_1$ in $B$ that is path-connected and evenly covered by $p$. Now the inverse image of $U$ by $p$ consists of slices $\{V_a\}$. Take a point $x$ in $E$ such that $p(x)=y$ and let $V'$ be the slice containing $x$.
Now the questions is, if $N$ is an arbitrary neighborhood of $x$ in $X$, then how to replace $U$ by a smaller neighborhood of $y$ such that the slice $V'$ containing $x$ can be included in the $N$? It seems very nontrivial to me...In the text I read, there is no detailed explanation about this. Could anyone give me some rigorous proof?