Many designers and User Interface (UI) educators discuss principles to be followed when designing the functional aspects of a UI. However, many UI principles have been proposed, scattered in scientific papers and teaching books. Some principles are different, and many are somehow similar or overlapping with others. This makes it very difficult to comprehend where to pay attention to when designing a UI. In this paper, we perform a systematic literature review to first identify the most relevant authors in the domain of functional UI design principles. Focusing on the three most cited works of these authors, we extracted 257 principles. We next analyzed all these principles, unified their variants, and, considering their scientific influence, finally derived a shorter and core selection of 36 principles. This core selection provides educators and UI designers with a clear path to teach, evaluate, learn and improve the UI functional design.