«زَيْدٌ الْأَمِيرُ»
Zayd is the emir — in the appropriate context, Zayd alone is the emir.
When the definite predicate is understood with generic al, the utterance may restrict the category or position of emirship to Zayd. If he truly is the only emir, this can be interpreted as a factual restriction.
But not every definite article functions in that way. If «الأمير» refers to a previously identified officeholder, the sentence may instead be about identifying that known individual. One must distinguish generic reference from anaphoric reference before inferring exclusivity.