Function images preserve subset ordering

up:: Image of Function

Let $f:X\to Y$ be a function, and $U\subseteq V\subseteq X$.

Let $y\in f(U)$. Then

$$y\in f(U)⟺\exists x\in U\mid f(x)=y\stackrel{U\subseteq V}{⟹}\exists x\in V\mid f(x)=y⟺y\in f(V)$$

Thus, images preserve subset ordering.