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Can I prevent a permanent from becomeing the target of a spell/ability by giving it hexproof/shroud in response?

Say, I have Illusionary Servant in play and the opponent targets it with the ability of Benalish Trapper. Can I prevent it from becoming a target and thus being destroyed by playing Plaxmanta in response? In other words, does a permanent become the target of an ability once the target is declared and the ability […]