The closure fuzzy mapping is employed to analyze approximation capability of feedforward regular fuzzy neural networks (FNNs) with two hidden layers. The bridge to do that is the fuzzy-valued Bernstein polynomial. By the fuzzy polynomial approximation theorem, this paper establishes some equivalent conditions for the fuzzy functions that can be approximated by the regular FNNs to any degree of accuracy. Thus, universal approximation of regular FNNs based on Zadeh's extension principle and fuzzy arithmetic can be studied, systematically. Finally, a simulation example is used to demonstrate the constructive procedure of approximating FNNs.