I don't understand your thought process. On the river, when you check-shove, I'm not sure what hand you're trying to rep.
If you had AK/AQ type hand I don't see you ever slowing down on the turn.
Preflop, he cold-called your 3 bet with the initial raiser left to act. I would never expect him to show up with A5/A6/66/55/65s or any 5x combo in this situation. I would expect his range to be something like 88-AA, A9-AQs, A10-AKo, KQs. I would discount some of those hands based on timing - also expect some of those hands to cold four bet sometimes.
Flop - Your bet seems standard. His call on the flop means that he has something either an ace or flush draw. I expect him to fold out all hands with no pair, no draws, and all pocket pairs that didn't hit as there's an ace on the flop and there's a player left to act. It also matters what suit your ace is. If you have the club that blocks all the nut flush draws...and he can only have KQcc maybe KJcc if he doesn't have an ace. So I expect his range to be weighted towards Ax.
Turn - the turn card doesn't really change anything...unless he has a flush draw and is worried about you filling up.
River - your check-shove is questionable to me. I'm not sure what you're trying to rep. Say you had a full house or quads....I don't think you would ever slow down on the turn unless it was precisely AA. If you had 66 or 55, you would expect your opponent to have an ace here a lot of the time...and should bet it for value on turn and river.
Your river check-shove is trying to fold out Ax hands, but your line makes your hand look like nuts/air. The only nut hand that you can conceivably play this way is AA...that makes it rather difficult for your opponent to fold when there is only one combo of AA left.
There's a lot of bluffs in your range like Mid-High pocket pairs...than value hands. Your line would be really good if you thought your opponent would be willing to value bet a hand like JJ on the river and call off thinking you're bluffing (I don't see how JJ gets to the river when it was 3 to the flop in a 3 bet pot with an ace on board though - your line is better if this hand was played heads up or if it was the initial raiser that flatted you preflop). If you've seen him make hero calls before....I wouldn't mind this play as you would be merging your hand.
As played, there's no way he's folding the river and that's why you got snapped and that's probably why you should have just check-called.
betrthanphil;253263 wroteI had A10c in this hand and thought that my play would make him fold an Ace here hoping to get him off an AJ or maybe even the same hand as me. I never wanted to get a call here thought id be behind. But I figured it was a good chance to try to get chips with so much in the pot already. He ended up snapping me.....I thought for sure I had lost when he flips over A10 too. I was shocked that he snapped me with a10.