Mapping is Surjection if its Direct Image Mapping is Surjection
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Theorem
Let $f: S \to T$ be a mapping.
Let $f^\to: \powerset S \to \powerset T$ be the direct image mapping of $f$.
Let $f^\to$ be a surjection.
Then $f: S \to T$ is also a surjection.
Proof
Let $y \in S$.
Since $f^\to$ is a surjection:
- $\exists U \in \powerset S: \map {f^\to} U = \set y$
From definition of direct image mapping:
- $\set y \ne \O \implies U \ne \O$
Let $x \in U$.
Then from definition of direct image mapping:
- $\map f x = y$
Since $y$ is arbitrary, $f$ is a surjection.
$\blacksquare$