I nominate Ehresmann's theorem according to which a proper submersion between manifolds is automatically a locally trivial bundle. It is incredibly useful, in deformation theory for example, but is sadly neglected in introductory courses and books on manifolds. It is completely elementary: witness [these lecture notes][1] by Peter Petersen, where it is proved in a few lines on page 9, the prerequisites being about two pages long.



[1]: http://www.math.ucla.edu/~petersen/manifolds.pdf