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 $A$.
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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
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Aiming for a contradiction, suppose $f \sqbrk S$ 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.
$\blacksquare$