[pilchard]

As always, it depends on the player but I would take the following factors into account.

He limps from EP and you raise so he should (depending on your table image) put you on a decent holding.

It is checked to you on the flop and you bet an AA7 r board. He calls. There are no draws there at all. Depending on the player he is unlikely to have a 7 after limping from EP. It all depends on your table image. If you make this isolation raise with a lot of hands (which you probably do given your 55 raise and the fact both blinds called you) he may think he is drawing live with a hand like KQ or KJ. The point is that he does not believe that you have an A.

The turn is a blank and he check calls. This situation is so specific to each other's playing characteristics. If you are the kind of player who keeps betting but are prepared to fold to a turn check raise he may have an A (although the river card reduces the chances of this).

On the river the 3rd A falls and it is checked to you. I would check here. As stated previously, he did not believe you had an A before the river card and he definitely doesn't now. Given he did not raise up front he is unlikely to have TT-KK. 99,88,66,44(unlikely as no turn raise),33,22 are possibilities. He may have paired a 7 instead. By checking you also remove the possibility of getting check raised as you would have to fold unless the player was extremely tricky.

He has shown no intention of folding a better hand. You do have a pair to showdown. If you bet 33 or 22 may call you depending on the player. You may get a K to call you but there are too few worse hands that will call to merit a bet in this position for my liking and you remove the possibility of being moved off your hand by a bluff raise.