The following is from page 14 of the book heat kernels and Dirac operators
Definition: let $\pi: \mathcal{E} \rightarrow M $ be a smooth map from a manifold $\mathcal{E}$ to a manifold $M$. We say that $(\mathcal{E},\pi)$ is a fiber bundle with typical fiber $E$ over$M$ of there is a covering of $M$ by open sets $U_i$ and diffeomorphisms $\phi_i : \pi^{-1} (U_i) \rightarrow U_i \times E,$ such that $\pi: \pi^{-1}(U_i) \rightarrow U_i$ is the decomposition of $\phi_i$ with projection onto the first factor $U_i$ in $U_i \times E$. The space $\mathcal{E}$ is called the total space of the fiber bundle, and $M$ is called the base.
It follows from the definition that $\pi^{-1}(x)$ is diffeomorphic to $E$ for all $x \in M$.
My goal is to prove the last statement.
Let $x$ be in some $U_i$ in $M$. Since $\pi_i$ is the decomposition of $\phi_i$ with projection onto the first factor $U_i$ in $U_i \times E$. Then $\phi_i$ restricted to $\pi^{-1}(x)$ has an image in $\lbrace x \rbrace\times E$. Since $\phi$ is injective then its restriction to $\pi^{-1}(x)$ is also injective. But I'm not able to conclude why it's surjective ?
