The logic KD45(C} (where C stands for a recursively axiomatized fuzzy propositional logic extending the basic logic BL) was studied by Hajek. In addition, Hajek studies the fuzzy modal logic S5(C) which is proved that formulae of S5(C) are in the obvious one-one isomorphic correspondence with formulae of the monadic fuzzy predicate calculus with unary predicates and just one object variable x. The correspondence maps tautologies of the modal logic to tautologies of the monadic predicate logic and the same for standard tautologies. By using this translation, Hajek is able to give an axiomatization for S5(C) and is also able to obtain several results on arithmetical complexity concerning these logics. In this presentation, we are going to introduce a faithful translation that preserves and reflects provability (or validity). According to this translation, we are also able to give an indirect axiomatization for KD45(C). We are going to focus our presentation in showing all these results.