Horizontal Section preserves Subsets

Theorem

Let $X$ and $Y$ be sets.

Let $A \subseteq B$ be subsets of $X \times Y$.

Let $y \in Y$.

Then:

$A^y \subseteq B^y$

where $A^y$ is the $y$-horizontal section of $A$ and $B^y$ is the $y$-horizontal section of $B$.

Proof

Note that if:

$x \in A^y$

from the definition of $x$-vertical section, we have:

$\tuple {x, y} \in A$

so:

$\tuple {x, y} \in B$

So, from the definition of $x$-vertical section, we have:

$x \in B^y$

So:

if $x \in A^y$ then $x \in B^y$.

That is:

$A^y \subseteq B^y$

$\blacksquare$