Proof-question: Pullback-bundle is a vector bundle

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I refer to the following question. In Creo's answer $\psi$ is used as the trivialization. I have problems verifying that $\psi$ indeed fulfills the conditions of a trivialization, especially that it is a bijection in the first place. The problem seems to be the injectivity of $\psi$ or rather of $\text{pr}_2 \circ \phi$. Why would the latter be injective?