I came across the definition of a fiber bundle in May's "A Concise Course in Algebraic Topology" (http://www.math.uchicago.edu/~may/CONCISE/ConciseRevised.pdf, Chapter 7, section 4). There were two main parts to the definition used. The first part mentioned homeomorphisms $ \phi: U \times F \rightarrow p^{-1}(U)$ such that $ p \phi = \pi_1$, where $F$ is a fixed topological space and and $U$ is an open set in a countable open cover of $E$.
After that, it is claimed that there are continuous maps $\lambda_U: B \rightarrow I$ such that $\lambda_U^{-1}(0, -1] = U$ and the cover is locally finite.
Are the conditions in the second part of the definition implied by the first, or are they just additional conditions for a "well chosen" open cover? If they are implied by the first part, why is that the case?