  We prove a version of the implicit function theorem for Lipschitz mappings
$f:\mathbb{R}^{n+m}\supset A \to X$ into arbitrary metric spaces. As long as
the pull-back of the Hausdorff content $\mathcal{H}_{\infty}^n$ by $f$ has
positive upper $n$-density on a set of positive Lebesgue measure, then, there
is a local diffeomorphism $G$ in $\mathbb{R}^{n+m}$ and a Lipschitz map
$\pi:X\to \mathbb{R}^n$ such that $\pi\circ f\circ G^{-1}$, when restricted to
a certain subset of $A$ of positive measure, is a the orthogonal projection of
$\mathbb{R}^{n+m}$ onto the first $n$-coordinates. This may be seen as a
qualitative version of a similar result of Azzam and Schul. The main tool in
our proof is the metric change of variables introduced in a paper of Hajlasz
and Malekzadeh.
