# Equivalence of Subobjects is Equivalence

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## 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.

Although this article appears correct, it's inelegant. There has to be a better way of doing it.In particular: That's not a relation since we can't know we're dealing with sets; cover thisYou can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by redesigning it.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{Improve}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## 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.

$\blacksquare$