Image of Set under Mapping is Set
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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Aiming for a contradiction, suppose $f \sqbrk S$ is not a set.
Then $f \sqbrk S$ must be proper.
But this contradicts Surjection from Class to Proper Class.
Hence the result.