Order Isomorphism Preserves Initial Segments

Theorem
Let $A_1$ and $A_2$ be classes.

Let $\prec_1$ and $\prec_2$ be relations.

Let $\phi : A_1 \to A_2$ create an order isomorphism between $\left({ A_1, \prec_1 }\right)$ and $\left({ A_2 , \prec_2 }\right)$.

Suppose $x \in A_1$.

Then, $\phi$ maps the $\prec_1$-initial segment of $x$ to the $\prec_2$-initial segment of $\phi\left({ x }\right)$.

Proof
$\phi$ maps the $\prec_1$-initial segment of $x$ to:

This is the $\prec_2$-initial segment of $\phi\left({x}\right)$.