If I put damage enough on deploy the fleet to destroy it and it leaves play do I still get the capital ship discount? Isn't there a general rule on most of these LCG's that cards only interact with the game while still in play unless implicitly stated otherwise.

So If I put 5 damage on Deploy the Fleet and it leaves play that 6 cost destroyer I wanted to play for 1 resource in fact costs 6 ?

Given that Deploy the Fleet has language structured identically to Admiral's Orders, which is an Event card and thus discarded immediately after resolving, I would believe that they work the same way. That is, the Action of either card creates an effect independent of the card itself that lasts until you play a Capital Ship card, or the end of the phase. So your Capital Ship would cost 1 resource, not 6.

Ah yes makes sense. Thanks

Yep, Deploy the Fleet creates a lasting effect. It has to be in play to trigger its Action, but does not have to remain in play for the effect to persist.

Keep in mind that if you deal enough damage to Deploy the Fleet to destroy it but have not yet focused the objective to use its resource, then you can no longer spend that resource because the objective is destroyed before you have the chance to spend resources to deploy a capital ship. Make sure that you spend that resource on something before you willingly destroy the objective.