A common misconception for beginner players is that you can counter or stop an activated ability from resolving if you destroy or removed the source of the activated ability. This is not correct. Once an ability has been activated or triggered, it is on the stack and independent of the source that generated it.

For example, if the ability of a Prodigal Pyromancer is activated it does not matter if the Pyromancer is in play when the ability resolves. If the Pryomancer is destroyed by a Doom Blade after the ability is played but before it resolves, the Pyromancer ability will still resolve.

Somewhat counter-intuitively, when a player leaves the game all spells and abilities that they control will leave the stack. However this obviously only applies to multi-player games.