In this paper, we propose a new efficient algorithm called Dep-Miner for discovering minimal non-trivial functional dependencies from large databases. Based on theoretical foundations, our approach combines the discovery of minimal functional dependencies along with the construction of real-world Armstrong relations (without additional execution time). These relations are small Armstrong relations taking their values in the initial relation. Discovering both minimal functional dependencies and real-world Armstrong relations facilitate the tasks of database administrators when maintaining and analyzing existing databases. We evaluate Dep-Miner performances by using a benchmark database. Experimental results show both the efficiency of our approach compared to the best current algorithm (i.e. Tane) described in a detailed way in this article, and the usefulness of real-world Armstrong relations.