> **Definition.** Suppose \$$f : X \to Y \$$ is a function between sets. For any \$$S \subseteq X \$$ define

$$f_{!}(S) = \\{ y \in Y: y = f(x) \textrm{ for all } x \in S \\} .$$

> This is a subset of \$$Y \$$. People pronounce \$$f_{!}\$$ as "\$$f\$$ lower shriek".

I think there's a mistake in your definition of "\$$f\$$ lower shriek" - shouldn't it be "the \$$y\$$s such that every \$$x\$$ sent to \$$y\$$ is in \$$S\$$"?