Increasing Mapping Preserves Lower Bounds

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Let $L = \left({S, \preceq}\right)$, $L' = \left({S', \preceq'}\right)$ be ordered sets.

Let $f:S \to S'$ be an increasing mapping.

Let $x \in S$, $X \subseteq S$ such that

$x$ is lower bound for $X$.

Then $f \left({x}\right)$ is lower bound for $f \left[{X}\right]$.


Let $y \in f\left[{X}\right]$.

By definition of image of set:

$\exists z \in X: y = f \left({z}\right)$

By definition of lower bound:

$x \preceq z$

Thus by definition of increasing mapping:

$f \left({x}\right) \preceq' y$