This is from Topology by Munkres:
Let $p:E \to B$ be a covering map. Suppose $U$ is a open set of $B$ that is evenly covered by $p$. Show that if $U$ is connected, then the partition of $p^{-1}(U)$ into slices is unique.
What I've tried so far
I proved that if $\{V_\alpha\}$ is a slice then each $V_\alpha$ is connected, but I don't know what to do next.