Inverse Relation Equal iff Subset

Theorem
If a relation $\RR$ is a subset or superset of its inverse, then it equals its inverse.

That is, the following are equivalent:

$(1)$ implies $(2)$
Suppose $\RR \subseteq \RR^{-1}$.

Then:

Hence $\RR^{-1} \subseteq \RR$.

$(2)$ implies $(1)$
Since $(1)$ implies $(2)$ was already established above, interpreting it on $\RR^{-1}$ yields:


 * $\RR^{-1} \subseteq \paren {\RR^{-1} }^{-1}$

By Inverse of Inverse Relation, $\paren {\RR^{-1} }^{-1} = \RR$.

Hence $(2)$ implies $(1)$.

$(3)$ $(1)$ and $(2)$
This is precisely the definition of set equality.