Complement of Horizontal Section of Set is Horizontal Section of Complement

Theorem
Let $X$ and $Y$ be sets.

Let $E \subseteq X \times Y$.

Let $y \in Y$.

Then:


 * $\paren {\paren {X \times Y} \setminus E}^y = X \setminus E^y$

where:


 * $\paren {\paren {X \times Y} \setminus E}^y$ is the $y$-horizontal section of the set difference $\paren {X \times Y} \setminus E$
 * $E^y$ is the $y$-horizontal section of $E$.

Proof
Note that from the definition of set difference, we have that:


 * $x \in X \setminus E^y$




 * $x \in X$ and $x \not \in E^y$.

That is, from the definition of the $y$-horizontal section:


 * $x \in X$ and $\tuple {x, y} \not \in E$.

This is equivalent to:


 * $\tuple {x, y} \in \paren {X \times Y} \setminus E$

From the definition of the $y$-horizontal section, this is then equivalent to:


 * $x \in \paren {\paren {X \times Y} \setminus E}^y$

So we have:


 * $x \in X \setminus E^y$ $x \in \paren {\paren {X \times Y} \setminus E}^y$.

So:


 * $\paren {\paren {X \times Y} \setminus E}^y = X \setminus E^y$