Suppose $M$,$N$ are smooth manifolds. $f:M\to N$ is a $C^{\infty}$ mapping. Let: $$f^*TN=\bigcup\limits_{p\in M}\{p\}\times T_{f(p)}N,$$ then how to prove that $f^*TN$ is a vector bundle with rank $n=\dim N?$
I think the first step is to prove that $f^*TN$ is a smooth manifold, but I can't easily find out the differential structure. The second step is to find out the local trivialization, I think it would be $\psi_{\alpha}: U_\alpha\times \mathbb{R}^{n}\to \pi^{-1}U_\alpha$ $$\psi_{\alpha}\left(p, v\right)=\left(p,\left.v^{\lambda} \frac{\partial}{\partial y_{\alpha}^{\lambda}}\right|_{f(p)}\right), \forall p \in U_{\alpha},\left(v\right) \in \mathbb{R}^{n},$$ here $U_\alpha$ is open covering of $M$ and $y_\alpha$ is local coordinates of $N.$