Definition:Image of Class under Mapping/Warning

Image of Class under Mapping: Warning
Let $V$ be a basic universe

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

Let $A$ be a class. Let $x \in A$ be a set of sets

Then:
 * $f \sqbrk x$ is not necessarily the same as $\map f x$

where:
 * $f \sqbrk x$ denotes the image of $x$ under $f$ where $x$ is treated as a class
 * $\map f x$ denotes the image of $x$ under $f$ where $x$ is treated as an element of the class $A$.

Thus:
 * $\map f x$ is what you get by applying $f$ to $x$
 * $f \sqbrk x$ is what you get by applying $f$ to each of the elements of $x$ (but not $x$ itself) and then gathering the results into a set.

Proof
Consider the mapping $f$ defined on the class of all ordinals $\On$ as:
 * $\forall \alpha \in \On: \map f \alpha = 5 \times \alpha$

where $\times$ denotes ordinal multiplication.

Consider the ordinal $4$

We have that:
 * $\map f 4 = 20$

but:
 * $f \sqbrk 4 = \set {5 \times \beta: \beta \in 4}$

From the von Neumann construction of natural numbers:
 * $4 = \set {0, 1, 2, 3}$

and so:
 * $f \sqbrk 4 = \set {0, 5, 10, 15}$

As can be seen, they are not the same thing at all.