I'm reading the Davey and Priestley's Introduction to Lattices and Order. On page 3, it is said that order-isomorphism is necessarily bijective. I thought we could easily prove the claim given the definition of order-isomorphism. However the book gave the following proof (the function $\phi$ is a function from set $P$ onto set $Q$ such that $x \le $y iff $\phi(x) \le \phi(y)$:
\begin{align*} \phi(x) = \phi(y) &\Leftrightarrow \phi(x) \le \phi(y) \;\&\; \phi(y) \le \phi(x) \\ &\Leftrightarrow x \le y \;\&\; y \le x \\ &\Leftrightarrow x = y \end{align*}
Could anyone please help me understand why this proves that order-isomorphism is bijective? Is it because in a set, any element is equal to itself, which implies the function $\phi$ is injective?
Thanks,