Direct Image Mapping of Mapping is Mapping
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Theorem
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.
Proof
$f$, being a mapping, is also a relation.
Hence Direct Image Mapping of Relation is Mapping can be applied directly.
$\blacksquare$
Sources
- 1975: T.S. Blyth: Set Theory and Abstract Algebra ... (previous) ... (next): $\S 5$. Induced mappings; composition; injections; surjections; bijections