I tried to prove the following claim, can you tell me if my proof is correct, please? Thank you!
Claim: If $\langle X, \prec_X \rangle$ and $\langle Y, \prec_Y \rangle$ are isomorphic strict partial orders then they have the same Mostowski collapse.
Proof: Let $f: X \to Y$ be an order isomorphism. Let $F: X \to \alpha$ and $G : Y \to \beta$ be the respective collapsing functions. Then $G \circ f : X \to \beta$ is also a collapsing function. By the uniqueness of the Mostowski collapse it follows that $\alpha = \beta$.