A stronger form of the theorem constructing a rigid binary relation on any set

On every set A there is a rigid binary relation i.e. such a relation R \subseteq A \times A that there is no homomorphism (A,R) \rightarrow (A,R) except the identity (Vop{\v{e}}nka et al. [1965]). We prove that for each infinite cardinal number \kappa if card A \leq 2^\kappa, then there exists a relation R \subseteq A \times A with the following property: \forall (x \in A) \exists ({x} \subseteq A(x) \subseteq A, card A(x) \leq \kappa) \forall (f: A(x) \rightarrow A, f \neq id_A(x)) f is not a homomorphism of R. The above property implies that R is rigid. If a relation R \subseteq A \times A has the above property, then card A \leq 2^\kappa.