Business rules play a critical role in building and maintaining effective and flexible information systems. In light of that critical role, the publication of the Semantic Business Vocabulary and Business Rules standard (SBVR), has been regarded a highly significant advance. Following that release, a number of research efforts have been made to convert SBVR to design models, most of which are structural models represented in UML. However, so far the proposed methodologies tend to be of an exploratory nature in the sense that they are not built on a rigorous foundation. Our aim is to identify a core subset of the SBVR features and show how those core SBVR features can be translated into an equivalent UML structural model. To do that on a sound foundation, we first provide formal models of the core SBVR and the target UML class diagram. We then transform the core SBVR model to the UML class model, completed with proofs of correctness, and describe how the mapping rules can be applied in a transformation process. Finally, to show the usefulness of our formal approach, we discuss how it is used as a crucial component in a larger project, which embraces a number of practical objectives.