Order Isomorphism Preserves Initial Segments

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

Let $\prec_1$ and $\prec_2$ be strict orderings.

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

Suppose $x \in A_1$.

Then $\phi$ maps the $\prec_1$-initial segment of $x$ to the $\prec_2$-initial segment of $\map \phi x$.

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

This is the $\prec_2$-initial segment of $\map \phi x$.