Local submersion theorem

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Statement

Let M and N be differential manifolds of dimensions m > n. Let f: M \to N be a differentiable map from M to N. Let p be a point such that f is a submersion at p, i.e. the induced map on tangent spaces is surjective at p. Then, there exists a neighbourhood U \ni p in M and V \ni f(p) in N and coordinate charts for U and V such that in those coordinate charts, the map is a projection map:

(x_1, x_2, \ldots, x_m) \mapsto (x_1, x_2, \ldots, x_n)