Equivalence of Subobjects is Equivalence
Theorem
Let $\mathbf C$ be a metacategory.
Let $C$ be an object of $\mathbf C$.
Let $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ be the category of subobjects of $C$.
The relation $\sim$ on $\mathbf{Sub}_{\mathbf C} \left({C}\right)$ defined by:
- $m \sim m'$ iff $m$ and $m'$ are equivalent
is an equivalence.
That's not a relation since we can't know we're dealing with sets; cover this
You can help Proof Wiki by redesigning it.
To discuss this proof in more detail, feel free to use the talk page.
If you are able to fix this proof, then when you have done so you can remove this instance of {{Improve}} from the code.
If you would welcome a second opinion as to whether your improvement is correct, add an invitation to {{Proofread}} the page (see the proofread template for usage).
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.
$\blacksquare$
