Equivalence of Subobjects is Equivalence
Theorem
Let $\mathbf C$ be a metacategory.
Let $C$ be an object of $\mathbf C$.
Let $\map {\mathbf {Sub}_{\mathbf C} } C$ be the category of subobjects of $C$.
The relation $\sim$ on $\map {\mathbf {Sub}_{\mathbf C} } C$ defined by:
- $m \sim m'$ if and only if $m$ and $m'$ are equivalent
is an equivalence.
That's not a relation since we can't know we're dealing with sets; cover this
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Proof
By Inclusion Relation on Subobjects is Preordering, $\subseteq$ is a preordering.
From Preorder Induces Equivalence, we see that $\sim$ is the equivalence induced by $\subseteq$.
In particular, $\sim$ is an equivalence.
$\blacksquare$