Existence and Uniqueness of Image of Relation
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Theorem
Let $V$ be a basic universe.
Let $\RR \subseteq V \times V$ be a relation.
Then the image $\Img \RR$ of $\RR$ exists and is unique.
Proof
By the Axiom of Specification we can create the subclass of $V$:
- $\Img \RR = \set {y \in V: \exists x \in V: \tuple {x, y} \in \RR}$
Hence $\Img \RR$ exists.
Suppose $\QQ \subseteq V$ such that $\QQ$ and $\Img \RR$ are both the image of $\RR$.
Then:
- $\QQ = \set {y \in V: \exists x \in V: \tuple {x, y} \in \RR}$
Thus:
- $x \in \QQ \implies x \in \Img \RR$
and:
- $x \in \Img \RR \implies x \in \QQ$
Hence by the Axiom of Extension:
- $\QQ = \Img \RR$
and uniqueness has been demonstrated.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $2$: Some Basics of Class-Set Theory: $\S 8$ Relations