We was proved that geodesic and holomorphically projective mappings of (pseudo-) Riemannian manifolds preserve the class of differentiability. Also, if the Einstein space admits a nontrivial geodesic mapping onto a (pseudo-) Riemannian manifold V, then V is an Einstein space. If a four-dimensional Einstein space with non-constant curvature globally admits a geodesic mapping f onto a (pseudo-) Riemannian manifold V with differentiability metric, then the mapping f is affine and, moreover, if the scalar curvature is non-vanishing, then the mapping is homothetic.