Image of Set under Mapping is Set

Theorem
Let $A$ be a class.

Let $\mathrm U$ denote the universal class.

Let $f: A \to \mathrm U$ be a class mapping.

Let $S$ be a subset of $A$.

Then the image $f \sqbrk S$ is also a set.

If $A$ is a set, then this result is known as the Axiom of Replacement in Zermelo-Fraenkel set theory.

Proof
Aiming for contradiction, suppose that $f \sqbrk S$ is not a set.

Then $f \sqbrk S$ must be proper.

By Restriction of Mapping to Image is Surjection, the restriction $f \restriction_{S \times f \sqbrk S}$ is a surjection.

But this contradicts Surjection from Class to Proper Class.

Thus by contradiction, $f \sqbrk S$ is a set.

Hence the result.