Preimage of Horizontal Section of Function is Horizontal Section of Preimage

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

Let $f : X \times Y \to \overline \R$ be an extended real-valued function.

Let $y \in Y$.

Let $D \subseteq \R$.

Then:


 * $\paren {f^y}^{-1} \sqbrk D = \paren {f^{-1} \sqbrk D}^y$

where:
 * $f^y$ is the $y$-horizontal section of $f$
 * $\paren {f^{-1} \sqbrk D}^y$ is the $y$-horizontal section of $f^{-1} \sqbrk D$.

Proof
Note that:


 * $x \in \paren {f^y}^{-1} \sqbrk D$




 * $\map {f^y} x \in D$

from the definition of preimage.

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


 * $\map f {x, y} \in D$

From the definition of preimage, this is equivalent to:


 * $\tuple {x, y} \in f^{-1} \sqbrk D$

Which in turn is equivalent to:


 * $x \in \paren {f^{-1} \sqbrk D}^y$

from the definition of the $y$-horizontal section.

So:


 * $x \in \paren {f^y}^{-1} \sqbrk D$ $x \in \paren {f^{-1} \sqbrk D}^y$.

giving:


 * $\paren {f^y}^{-1} \sqbrk D = \paren {f^{-1} \sqbrk D}^y$