I do not really understand what to do with the following problem and would appreciate any help.
Let $\Bbb{Z}=\{[0]_6,[1]_6,[2]_6,[3]_6,[4]_6,[5]_6\}. $Consider the purported function $f:\Bbb{Z}_6\rightarrow\Bbb{Z}$ defined by $f([a]_6)=a$ for all $[a]_6\in \Bbb{Z}_6$. Show that $f$ is not well defined.
Attempt:
Suppose for contradiction that the putative function $f$ is well defined and therefore single-valued.
By previously established lemma :
Let $m \gt 1$ be an integer.For all $a,b \in \Bbb{Z}$ the following holds:
1.) $[a]_m = [b]_m $ iff $a \equiv b \pmod m$
2.) $a \in [b]_m$ iff $a \equiv b \pmod m$
Since $0 \equiv 6 \pmod 6$ then by lemma (1), $[0 ]_6=[6]_6$. Applying the function on both sides yields $0=6$ which is a contradiction.
I am unsure because I thought $a$ and $b$ must be restricted to lie between $0$ and $5$ given the defintion of $\Bbb{Z}=\{[0]_6,[1]_6,[2]_6,[3]_6,[4]_6,[5]_6\}$