Show that If $C$ is a chain in $X$ then $f(C)$ is also chain in $Y$.

Question

Let X and Y are poset and $f:X\to Y$ is increasing function. If $C$ is a chain in $X$, show that $f(C)$ is also chain in $Y$.

Since C is chain for every $x,y \in C: (x,y)\to \left(x\leq y\bigvee y\leq x\right)$.

So $f(x)\leq f(y)$ or $f(y)\leq f(x)$ because f is increasing. Since $x,y\in C $ and $f(x),f(y)\in f(C)\Rightarrow \left(f(x)\leq f(y)\bigvee f(y)\leq f(x)\right)$ is this enough?

Answer 1

Suppose $y_1, y_2$ are in $f[C]$. Then there are $x_1,x_2 \in C$ such that $f(x_1) = y_1$ and $f(x_2) = y_2$. As $C$ is a chain, we have either $x_1 \le x_2$ or $x_2 \le x_1$ and as $f$ is increasing, we also have either $y_1 = f(x_1) \le f(x_2) = y_2$ or $y_2 = f(x_2) \le f(x_1) = y_1$, as required.