We prove that the property of a set-valued mapping
F
:
X
⇉
Y
F:X \rightrightarrows Y
to be locally metrically regular (and consequently, the properties of the mapping to be linearly open or pseudo-Lipschitz) is separably reducible by rich families of separable subspaces of
X
×
Y
X\times Y
. In fact, we prove that, moreover, this extends to computation of the functor
{reg}
F
\textrm {{reg}}\, F
that associates with
F
F
the rates of local metric regularity of
F
F
near points of its graph.