Talk:Image of Subset under Relation equals Union of Images of Elements

In the proof that $\mathcal R \left[{X}\right] \subseteq \bigcup_{x \in X} \mathcal R \left({x}\right)$, I found the usage of $x$ and $y$ in the following line a little bit confusing:

$\left({x, y}\right) \in \bigcup_{x \in X} \left\{{ \left({x, y}\right) \in \mathcal R }\right\}$

On the LHS of the $\in$, $y$ is an arbitrarily chosen value in $\mathcal R \left[{X}\right]$ and $x$ is the value we chose that relates to $y$ under $\mathcal R$. However, on the RHS of the $\in$, the letter $x$ is reused for iterating through all the values in $X$. Would you mind if I changed the $x$ to a $s$ and the $y$ to a $t$ on the LHS? The first few lines would read:

\(\ds \implies \ \ \)

\(\ds \exists s \in X: \left({s, t}\right)\)

\(\in\)

\(\ds \mathcal R\)

Definition of $\mathcal R \left[{X}\right]$

\(\ds \implies \ \ \)

\(\ds \left({s, t}\right)\)

\(\in\)

\(\ds \bigcup_{x \mathop \in X} \left\{ {\left({x, y}\right) \in \mathcal R}\right\}\)

Definition of Set Union

\(\ds \implies \ \ \)

\(\ds t\)

\(\in\)

\(\ds \bigcup_{x \mathop \in X} \left\{ {y \in T: \left({x, y}\right) \in \mathcal R}\right\}\)

Definition of Relation

--Cjhanrahan (talk) 04:00, 1 December 2017 (EST)

No, because what you have written does not seem to make any sense. --prime mover (talk) 13:51, 1 December 2017 (EST)

Haha ok, I'll leave it as is.--Cjhanrahan (talk) 04:01, 2 December 2017 (EST)