In nonlinear Model Predictive Control (MPC), an optimal control problem is solved at each time instant to make an educated decision on the next control input. We propose a framework to ensure desired properties of the resulting MPC closed loop if data data-driven surrogate models are leveraged in the optimization step. Moreover, we show that the imposed assumptions can be rigorously verified using Koopman theory. To be more precise, we employ extended dynamic mode decomposition (EDMD) in the prediction step, provide an in-depth analysis of the approximation error, and propose novel proportional error bounds.