Mapping whose Domain is Small Class is Small
Jump to navigation
Jump to search
Theorem
Let $F$ be a mapping.
Let the domain of $F$ be a small class.
Then, $F$ is a small class.
Proof
Let $A$ denote the domain of $F$.
Let $B$ denote the image of $F$.
Since $F$ is a mapping, $F$ is also a relation.
Therefore:
- $F \subseteq A \times B$
where $A \times B$ denotes the Cartesian product of $A$ and $B$.
$B$ is the image of $A$ under $F$ and is therefore a small class by Image of Small Class under Mapping is Small.
Since $A$ and $B$ are both small, their Cartesian Product is Small.
By Axiom of Subsets Equivalents, it follows that $F$ is small.
Sources
- 1971: Gaisi Takeuti and Wilson M. Zaring: Introduction to Axiomatic Set Theory: $\S 6.15$