Mapping/Mistakes/Image Elements not in Codomain

Example of Mistake in Definition of Mapping
This example of an attempted definition of a mapping contains a mistake.
 * $f: \N \to \N$ defined as: $\forall x \in \N: x \mapsto x - 7$

Explanation
The codomain of $f$ is:
 * $\Cdm f = \set {0, 1, 2, \ldots}$

However, consider the subset $S \subset \Dom f$ of the domain of $f$ where $S = \set {0, 1, 2, 3, 4, 5, 6}$:
 * $f \sqbrk S = \set {-7, -6, -5, -4, -3, -2, -1}$

and none of the elements of the image of $S$ are actually elements of $\Cdm f$.

That is:
 * $\Img f \nsubseteq \Cdm f$