I proved the following equality for $latex \mathcal{A}$ being a filter object and $latex f$ being either a funcoid: $latex \mathrm{dom}f|^{\mathsf{FCD}}_{\mathcal{A}} = \mathcal{A} \cap^{\mathfrak{F}} \mathrm{dom}f$ or reloid: $latex \mathrm{dom}f|^{\mathsf{RLD}}_{\mathcal{A}} = \mathcal{A} \cap^{\mathfrak{F}} \mathrm{dom}f$.

See the updated version of Funcoids and Reloids online article.

The proofs are short but they may be not as obvious at it seems.