Order Isomorphism is Surjective Order Embedding

Theorem
Let $\struct {S, \preceq_1}$ and $\struct {T, \preceq_2}$ be ordered sets.

Let $f: S \to T$ be a mapping.

Then $f$ is an order isomorphism :
 * $(1): \quad f$ is a surjection
 * $(2): \quad \forall x, y \in S: x \preceq_1 y \iff \map f x \preceq_2 \map f y$

That is, $f$ is an order embedding which is also a surjection.

Necessary Condition
Suppose $f$ is an order isomorphism.

Then by definition $f$ is a bijection and so a surjection.

Also by definition, $f$ is increasing, and so:
 * $\forall x, y \in S: x \preceq_1 y \implies \map f x \preceq_2 \map f y$

Also by definition $f^{-1}$ is also a bijection which is increasing, and so:
 * $\forall x, y \in S: \map f x \preceq_2 \map f y \implies x = \map {f^{-1} } {\map f x} \preceq_1 \map {f^{-1} } {\map f y} = y$

and so:
 * $\forall x, y \in S: x \preceq_1 y \iff \map f x \preceq_2 \map f y$

Sufficient Condition
Suppose $f: S \to T$ is a mapping such that:
 * $f$ is a surjection;
 * $\forall x, y \in S: x \preceq_1 y \iff \map f x \preceq_2 \map f y$

From Order Embedding is Injection we have that $f$ is an injection.

As, by hypothesis, it is also surjective, it follows that it is a bijection.

Now, suppose $a, b \in T$.

As $f$ is surjective:
 * $\exists x, y \in S: \map f x = a, \map f y = b$

As $f$ is bijective, then:
 * $x = \map {f^{-1} } a, y = \map {f^{-1} } b$

So by hypothesis:
 * $a \preceq_2 b \implies \map f x \preceq_2 \map f y \implies \map {f^{-1} } a = x \preceq_1 y = \map {f^{-1} } b$

Hence, by definition, $f$ is an order isomorphism.