Image of Class under Mapping is Image of Restriction of Mapping to Class

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Theorem

Let $V$ be a basic universe

Let $f: V \to V$ be a mapping.

Let $A$ be a class.

Let $f \sqbrk A$ denote the image of $A$ under $f$.


Then $f \sqbrk A$ is the image of the restriction of $f$ to $A$:

$f \sqbrk A = \Img {f {\restriction} A}$


Proof

By definition, $f {\restriction} A$ is class of all ordered pairs $\tuple {a, \map f a}$, where $a \in A$.

The result follows from Restriction of Mapping is Subset of Cartesian Product.

$\blacksquare$


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