I have to tell you that I think you are thinking of him in terms of him being much older than he is in regards to "correcting" the bragging and fishing for compliments. I want to URGE you to change your way of thinking.
He needs your compliments and he needs to brag right now. He is 5, not 10. He is internalizing what he is supposed to be doing in the household. How he is supposed to feel about contributing to the household, and how he should feel about himself for having done it. Before this, you did for him exclusively. He realizes that he is a bigger boy now because you are asking for him to do certain things, he feels the expectation from you to do things for himself and for you. And he is dealing with it in the best way he knows how - talking about it to himself and to you.
LET HIM.
So what if he beats you to the punch to praise him? Say to him "Yes, you DID help mommy! Thank you very much for cleaning up your spilled water. I really like how you thought to do that without me even telling you!"
If you do this instead of trying correct him, he will feel loved and respected and want to continue to grow and try. As his little mind matures, he will stop bragging and talking to himself about it naturally and you two will have more mature conversations about his contributions to the household.
If you try to squash this, he will not understand what he is doing wrong because he is not mature enough to understand. He will just know that you are unhappy with him, and he will grow to feel that he can't do anything to please you. Later he will not WANT to please you, and then you'll be fussing at him to pick up his plate, put his underwear in the hamper, etc.
Be HAPPY that he is doing this. There are 5 year olds everywhere not lifting a finger to help, or having mini-meltdowns when asked to. You are one lucky mom.
Dawn