Injection from Proper Class to Class

Theorem
Let $A$ be a class.

Let $P$ be a proper class.

Let $f: P \to A$ be an injection.

Then $A$ is proper.

Proof
$A$ is not proper.

Then it must be a set.

By Injection to Image is Bijection, it follows that the restriction $f \restriction_{P \times f \sqbrk P}$ is a bijection.

By the corollary of Class is Proper iff Bijection from Class to Proper Class, $f \sqbrk P$ is proper.

But since $f \sqbrk P \subseteq A$, this contradicts Subclass of Set is Set.

And so by contradiction, $A$ cannot be a set.

Therefore $A$ is proper.