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.

$\blacksquare$