Horizontal Section preserves Increasing Sequences of Sets

Theorem

Let $X$ and $Y$ be sets.

Let $\sequence {A_n}_{n \mathop \in \N}$ be an increasing sequence in $X \times Y$.

Let $y \in Y$.

Then:

$\sequence {\paren {A_n}^y}_{n \mathop \in \N}$ is an increasing sequence.

Proof

Since $\sequence {A_n}_{n \mathop \in \N}$ is increasing, we have:

$A_n \subseteq A_{n + 1}$

for each $n$.

From Horizontal Section preserves Subsets, we have:

$\paren {A_n}^y \subseteq \paren {A_{n + 1} }^y$

for each $n$.

So:

$\sequence {\paren {A_n}^y}_{n \mathop \in \N}$ is an increasing sequence.

$\blacksquare$