Set $A$ and $B$ partially ordered sets, let $f: A \to B$ an isomorphism. Prove that:
If $C$ is a connected subset of $A$, then $ f[C] $ is a connected subset of $B$.
Definition connected subset: a subset $C$ of partially ordered set is connected if satisfies the next condition
If $a,b \in C $ and $a \leq x\leq b$, then $x \in C$
Now, my problem is that I don't know where I start, I understand that is a direct proof $p \Rightarrow q$. I can identify my hypothesis. I was thinking this form but I'm not sure let $a,b \in f[C]$ and suppose that $a \leq x \leq b$ $$\Rightarrow$$ $$\Rightarrow$$ $$\Rightarrow$$
I really appreciate that someone can help me with the detailed explanation to be able to identify where I should start the proof
Hints: You start with elements $a,b\in f(C)$ and assume there is some $x$ such that $a\leq x\leq b$. Now use that $f$ is an isomorphism, which means it is a bijection, so you can find preimages of each element. Then recall that the isomorphism preserves ordering which should let you use the connectivity of C. Finally push back forward into f(C)