Image of Union of Nest of Mappings is Union of Class of Images/Proof

Proof
From Union of Nest of Mappings is Mapping we have that $\bigcup N$ is a mapping.

Let $y \in \Img {\bigcup N}$.

Then by definition of mapping:
 * $\exists \tuple {x, y} \in \bigcup N$

Then by definition of union of class:
 * $\exists f \subseteq \bigcup N: \tuple {x, y} \in f$

Hence:
 * $\exists f \subseteq \bigcup N: y \in \Img f$

That is:
 * $y \in \ds \bigcup_{f \mathop \in N} \Img f$

That is:
 * $\Img {\bigcup N} \subseteq \ds \bigcup_{f \mathop \in N} \Img f$

Let $y \in \ds \bigcup_{f \mathop \in N} \Img f$.

Then:
 * $\exists f \subseteq \bigcup N: y \in \Img f$

Then by definition of mapping:
 * $\exists f \subseteq \bigcup N: \tuple {x, y} \in f$

Thus by definition of union of class:
 * $\exists \tuple {x, y} \in \bigcup N$

It follows that:
 * $y \in \Img {\bigcup N}$

That is:
 * $\ds \bigcup_{f \mathop \in N} \Img f \subseteq \Img {\bigcup N}$

Hence by definition of set equality:


 * $\Img {\bigcup N} = \ds \bigcup_{f \mathop \in N} \Img f$