How did the coat of arms of the Byzantine emperors become the coat of arms of Russia?

As a result of the marriage of the niece of the last Byzantine emperor Sophia Palaeologus with Ivan 3, Russia got the well-known coat of arms. Double-headed eagle. At that time, Byzantium was in great decline. The constant attacks of the Turks forced them to take retaliatory actions. Beyond the decline and danger of being conquered, this marriage could not have taken place. Similarly, the Ecumenical Council could not have taken place. But the need for an ally made adjustments. The marriage was contracted. An ally is chosen on the basis of faith in Christ, strength and geographic location.