This letter introduces a set-theoretic state estimation approach for bounded-error nonlinear discrete-time systems, subject to nonlinear observations or constraints, when polytope-valued uncertainties are assumed. Our approach relies on finding a polytopic enclosure to the true range of nonlinear mappings via the direct use of hyperplane and vertex representations of polytopes. In particular, we derive a tractable enclosure of the set-product of an interval and a polytope, which is then used in a two-step state estimation approach consisting of (i) state propagation (prediction) using the nonlinear system dynamics and (ii) measurement update (refinement) based on nonlinear observations. Moreover, we analyze the computational complexity of our proposed technique and derive sufficient conditions for stability of the estimation errors. Finally, we compare the effectiveness of our approach with existing polytopic and interval observers in the literature.