Restriction of Mapping to Small Class is Small

Theorem

Let $F$ be a mapping.

Let $A$ be a small class.

Then the restriction $F {\restriction_A}$ is a small class.

Proof

The domain of $F {\restriction_A}$ is a subset of $A$.

By Axiom of Subsets Equivalents, the domain is a small class.

By Mapping whose Domain is Small Class is Small, it follows that $F {\restriction_A}$ is a small class.

$\blacksquare$

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.16$