Vertical Section preserves Subsets

Theorem

Let $X$ and $Y$ be sets.

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

Let $x \in X$.

Then:

$A_x \subseteq B_x$

where $A_x$ is the $x$-vertical section of $A$ and $B_x$ is the $x$-vertical section of $B$.

Proof

Note that if:

$y \in A_x$

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:

$y \in B_x$

So:

if $y \in A_x$ then $y \in B_x$.

That is:

$A_x \subseteq B_x$

$\blacksquare$