Value of Relation is Small

Theorem

The value of a relation is always a small class.

Proof

Let $\RR$ be an arbitrary relation.

Let $s$ be any set.

The value of a relation is either equal to some set $y$ or $\O$ by Uniqueness Condition for Relation Value.

If it is equal to some set $y$, then the value of $s$ under $\RR$ is a small class by the definition of small class.

If it is equal to $\O$, then the result follows from Empty Set is Small.

Sources

• 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.13$