The main goal of this paper is to establish the convergence of mimetic discretizations of the first-order system that describes linear diffusion. Specifically, mimetic discretizations based on the support-operators methodology (SO) have been applied successfully in a number of application areas, including diffusion and electromagnetics. These discretizations have demonstrated excellent robustness, however, a rigorous convergence proof has been lacking. In this research, we prove convergence of the SO discretization for linear diffusion by first developing a connection of this mimetic discretization with Mixed Finite Element (MFE) methods. This connection facilitates the application of existing tools and error estimates from the finite element literature to establish convergence for the SO discretization. The convergence properties of the SO discretization are verified with numerical examples.