Restriction of Mapping to Small Class is Small
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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: $\S 6.16$