Axiom of Replacement implies Image of Bijection on Set is Set

Theorem
Let the Axiom of Replacement (in the context of class theory) be accepted.

Let $A$ be a class which can be put into one-to-one correspondence with a set.

Then $A$ is a set.

Proof
Let $x$ be a set such that $A$ can be put into one-to-one correspondence with $x$.

Then by definition there exists a bijection from $x$ to $A$.

Let $f: x \to A$ be such a bijection.

Then by the Axiom of Replacement, $f \sqbrk x$ is a set.

But:
 * $f \sqbrk x = A$

Hence the result.