The increasing demand for data quality technology has motivated revisions of classical dependencies to capture more inconsistencies in real-life data. A class of integrity constraints, referred to as functional dependencies with built-in predicates (PFDs), is proposed for relational databases and their axiomatization is investigated. In contrast to traditional functional dependencies (FDs) developed mainly for schema design, PFDs generalize the notions of FDs to apply to subsets of relations specified by constraints in the context of interpreted data, and aim at capturing the consistency of data by enforcing bindings of ranges of semantically related values. For the implication analysis of PFDs, which is to decide whether or not a set of PFDs entails another PFD, we provide an inference system analogous to Armstrong's axioms for FDs, and prove the soundness and completeness of the inference system. This work is a step towards a practical constraint-based method for improving data quality since inconsistencies and errors in databases often emerge as violations of integrity constraints.