Let $R \subset A^2$ be an equivalence class. Prove that $\bigcup_{a\in A}[a]=A$.
I have tried to prove it through double containment:
If $x\in A$, since $x\in [x]$, then $x\in \bigcup_{a\in A}[a]$. So $A\subset \bigcup_{a\in A}[a]$.
Now, if $x\in \bigcup_{a\in A}[a]$, then exists $[m]$, where $m \in A$, such that $x\in [m]$. But $x\in [m]$ is the case (by definition of equivalence class) if and only if $x\in A$ and $xRm$. So, $\bigcup_{a\in A}[a]\subset A$.
Therefore, since $A\subset \bigcup_{a\in A}[a]$ and $\bigcup_{a\in A}[a]\subset A$ we conclude that is $\bigcup_{a\in A}[a]=A$.
I want to know if my proof is correct, can anyone help me please? Thank you so much gentle people