Direct Image Mapping is Mapping
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Theorem
Direct Image Mapping of Relation is Mapping
Let $S$ and $T$ be sets.
Let $\RR \subseteq S \times T$ be a relation on $S \times T$.
Let $\RR^\to: \powerset S \to \powerset T$ be the direct image mapping of $\RR$:
- $\forall X \in \powerset S: \map {\RR^\to} X = \set {t \in T: \exists s \in X: \tuple {s, t} \in \RR}$
Then $\RR^\to$ is indeed a mapping.
Direct Image Mapping of Mapping is Mapping
Let $S$ and $T$ be sets.
Let $f: S \to T$ be a mapping on $S \times T$.
Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$:
- $\forall X \in \powerset S: \map {f^\to} X = \set {t \in T: \exists s \in X: \map f s = t}$
Then $f^\to$ is indeed a mapping.