Let $M$ be a smooth manifold. Suppose we are given an open cover ${U_\alpha}$ of $M$ and for $\alpha,\beta$ ; a smooth map $\tau_{\alpha\beta}\colon U_\alpha \cap U_\beta \to GL(k; R)$ satisfying the cocycle condition $\tau_{\alpha\beta}\tau_{\beta\gamma}=\tau_{\alpha\gamma}$, Show that there is a unique, up to a bundle isomorphism, smooth vector bundle $E$ of rank k over M with transition functions $\tau_{\alpha\beta}$.
This is problem 5-4 in John. Lee's intro to smooth manifolds. I kind of constructed the equivalence $(p,v)\sim(q,w)$ if $p=q$ and there is a $\tau_{\alpha\beta}$ with $\tau_{\alpha\beta}(v)=w$. But how to proceed onward?
This is not a homework problem.