Proof verification: If $f:A\rightarrow B$ and $S \subset T \subset A$, then $f(S) \subset f(T)$

Question

Proof. If $S\subset T$, then for all $s\in S$ there exists $t \in T$ such that $f(s)=f(t)$.

In other words, for all $s\in S$ there exists $b \in f(T)$ such that $f(s)=b$.

Therefore, all $f(s)$ is contained in $f(T)$ and thus $f(S)\subset f(T)$. (QED?)

---

Is it correct?

Answer 1

No, it is not..

You should take an arbitrary $y\in f(S)$, then there is $x\in S$ such that $f(x)=y$. But since $x\in T$ we have also $y\in f(T)$, so $f(S)\subseteq f(T)$