Recent results on receding horizon control of linear systems with state and input constraints have shown that the optimal receding horizon controller is piecewise affine and continuous, with the resulting value function being piecewise quadratic and continuously differentiable. The purpose of this paper is to exploit these results to show that the controller renders the closed-loop system globally input-to-state stable (ISS) when the open-loop system is stable, and locally input-to-state stable when the open-loop plant is unstable. While the result is simple in nature, it has interesting implications in utilizing constrained receding horizon scheme in a switching based supervisory control framework.