I was reading the notes on bundles "Differential Topology of Fiber Bundles" by Karl-Hermann Neeb and on the final pages (138-139) "Homotopy theory of bundles" the author author makes a statement that, if we denote the set of equivalence classes of smooth principal $G$-bundles over a smooth manifold $M$ by $\mathbf{Bun}(M, G)$, and the topological $G$-bundles over $M$ by $\mathbf{Bun}(M, G)_{top}$, then the natural map
$$\mathbf{Bun}(M, G) \rightarrow \mathbf{Bun}(M, G)_{top},$$ is a bijection. But this result does not seem so trivial to me. I would like some idea of the proof or some reference that presents this result.
Appreciate.