I'm wondering how to show the following: let $E$, $B\_1$ and $B\_2$ smooth manifolds. Suppose $\rho:E\rightarrow B\_2$ is a smooth vector bundle and $f:B\_1\rightarrow B\_2$ a smooth map. If we write $$f^\*E:=\{(b, e)\in B\_1\times E: \rho(e)=f(b)\},$$ how can I show $f^\*E$ is a smooth submanifold of $B\_1\times E$. Any help will be valuable, thanks...