Rectifiable sets and coarea formula for metric-valued mappings

We study Lipschitz mappings defined on an Hn-rectifiable metric space with values in an arbitrary metric space. We find necessary and sufficient conditions on the image and the preimage of a mapping for the validity of the coarea formula. As a consequence, we prove the coarea formula for some classes of mappings with Hk-σ-finite image. We also obtain a metric analog of the Implicit Function Theorem. All these results are extended to large classes of mappings with values in a metric space, including Sobolev mappings and BV-mappings.