Union of Horizontal Sections is Horizontal Section of Union

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

Let $\set {E_\alpha : \alpha \in A}$ be a set of subsets of $X \times Y$.

Let $y \in Y$.

Then:


 * $\ds \paren {\bigcup_{\alpha \in A} E_\alpha}^y = \bigcup_{\alpha \in A} \paren {E_\alpha}^y$

where:
 * $\ds \paren {\bigcup_{\alpha \in A} E_\alpha}^y$ is the $y$-horizontal section of $\ds \bigcup_{\alpha \in A} E_\alpha$
 * $\paren {E_\alpha}^y$ is the $y$-horizontal section of $E_\alpha$.

Proof
Note that:


 * $\ds x \in \bigcup_{\alpha \in A} \paren {E_\alpha}^y$




 * $x \in \paren {E_\alpha}^y$ for some $\alpha \in A$.

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


 * $\tuple {x, y} \in E_\alpha$ for some $\alpha \in A$.

This in turn is equivalent to:


 * $\ds \tuple {x, y} \in \bigcup_{\alpha \in A} E_\alpha$

Again applying the definition of the $y$-horizontal section, this is the case :


 * $\ds x \in \paren {\bigcup_{\alpha \in A} E_\alpha}^y$

So:


 * $\ds x \in \bigcup_{\alpha \in A} \paren {E_\alpha}^y$ $\ds x \in \paren {\bigcup_{\alpha \in A} E_\alpha}^y$

giving:


 * $\ds \paren {\bigcup_{\alpha \in A} E_\alpha}^y = \bigcup_{\alpha \in A} \paren {E_\alpha}^y$