Show that ($|F (A)| = |A|$ for every measurable $A \subseteq U$) $\iff$ ($|$det $dF (x)| = 1$ ∀x ∈ U$)

Question

Let $U \subseteq $ $R^n$ be an open set, and let $F: U \rightarrow R^n$ be a diffeomorphism.

Show that:

($|F (A)| = |A|$ for every measurable $A \subseteq U$) $\iff$ ($|$det $dF (x)| = 1$ $∀x \in U$).