# 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

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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.

$\blacksquare$

## Sources

- 1971: Gaisi Takeuti and Wilson M. Zaring:
*Introduction to Axiomatic Set Theory*: $\S 6.7$