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.
$\blacksquare$
Sources
- 2010: Raymond M. Smullyan and Melvin Fitting: Set Theory and the Continuum Problem (revised ed.) ... (previous) ... (next): Chapter $6$: Order Isomorphism and Transfinite Recursion: $\S 1$ A few preliminaries