DEF-RRST. Relation Restriction.
Let $R$ be a binary relation. If $A\subseteq\operatorname{dom}R$, the left-restriction of $R$ to $A$ is
\[{R\restriction_A} = \{(x,y)\in A\times\operatorname{ran}R:x\,R\,y\}.\]If $B\subseteq\operatorname{ran}R$, the right-restriction of $R$ to $B$ is
\[{R\restriction^A} = \{(x,y)\in\operatorname{dom}R\times B:x\,R\,y\}.\]
PROP-RRST-EMP.
For any binary relation $R$,
\[{R\restriction_\varnothing} = {R\restriction^\varnothing} = \varnothing.\]