Injectivity of projection fibriation map of Hopf-like fibration on cohomology level

Question

Let we have fiber bundle $\xi$ with projection map $p : E \to X$. And let $\mathbb{P} \xi$ is projectivization of that fiber bundle: every fiber $E_p$ we change on fiber $\mathbb{P}E_p$. So I have map $f: \mathbb{P}E \to X$. Is it true that map $f^* : H^i(X) \to H^i(\mathbb{P}E)$ is injective?