How are bitcoins reassigned, that is, what about a transaction makes a "coin" no longer usable by the sender, and makes it usable by the recipient?
Alice wants to send bitcoins to Bob. Alice's wallet gathers transactions originally sent to Alice so that the monetary amount of said transactions is at least the amount that Alice wants to send Bob. The transaction is basically a message, so a little bit of ECDSA on top of that and boom, there's a transaction: only Alice could have created that particular transaction (message), and anyone can verify Alice's signature for that transaction.
Now Bob "receives" the money. Why can he use it? (Indeed, why could Alice use the money above...) Digital signatures of themselves don't solve this problem, i.e. the transaction/message along with its signature doesn't "transfer" anything, there's gotta be some other ingredient involved, but what is it? I can't see, or at least not understand, the answer in the various sources I've read.
It has to be something about Bob's private key, but I can't see how it's used to "unlock" the money he received from Alice. I don't know what transactions look like exactly, but suppose they're basically of the form tx_1 = (Alice's public key)+amount+(Bob's public key) - would Bob's wallet basically be saying "reproduce Bob's public key and you can access tx_1"? I guess that would do the trick, but then again Bob was already "logged on" to his wallet so he's already proven the ability to reproduce Bob's public key.
Ok I think I'm getting ahead of myself. Anyway, received transactions, how are they "unlocked"? Thanks.
(Yes, I have tried finding an answer on BSE and elsewhere already.)