Composition of Direct Image Mappings of Mappings/Proof 2
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Theorem
Let $A, B, C$ be non-empty sets.
Let $f: A \to B$ and $g: B \to C$ be mappings.
Let:
- $f^\to: \powerset A \to \powerset B$
and
- $g^\to: \powerset B \to \powerset C$
be the direct image mappings of $f$ and $g$.
Then:
- $\paren {g \circ f}^\to = g^\to \circ f^\to$
Proof
We have that a mapping is a relation.
Hence Composition of Direct Image Mappings of Relations applies.
$\blacksquare$