Preimages preserve subset ordering

up:: Preimage of Function

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

Let $x\in f^{-1}(U)$. Then

$$x\in f^{-1}(U)⟺f(x)\in U\stackrel{U\subseteq V}{⟹}f(x)\in V⟺x\in f^{-1}(V)$$

Thus, preimages preserve subset ordering.