Image of Small Class under Mapping is Small

Theorem
Let $A$ be a mapping.

Let $a$ be a small class.

Then, the image of $a$ under $A$ is small.

Proof
Since $A$ is a mapping:


 * $\forall y: \exists x: \forall z: \left({ y A z \implies z = x }\right)$

This satisfies the antecedent of the axiom of replacement. Therefore:


 * $\forall w: \exists x: \forall y: \left({ y \in w \implies \forall z: \left({ y A z \implies z \in x }\right) }\right)$

Universal Instantiation yields:


 * $\exists x: \forall y: \left({ y \in a \implies \forall z: \left({ y A z \implies z \in x }\right) }\right)$

By applying the definition for the restricted universal quantifier and rearranging quantifiers:


 * $\exists x: \forall z: \left({ \exists y \in a: y A z \implies z \in x }\right)$

Applying the definition for image:


 * $\exists x: \operatorname{Im} \left({a}\right) \subseteq x$

By Axiom of Subsets Equivalents, the image of $a$ under $A$ must be small.