Does my fully charged AirPod 2 pro case recharge my airpods

Yes.

Charge AirPods, AirPods Pro, or AirPods Max - Apple Support

Of course! :-)

"Does my fully charged AirPod 2 pro case recharge my airpods: Does my fully charged case recharge my AirPods after I use them?"

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Charing Wirelessly:

You'd need a MagSafe Charger.

1. Go here: Identify your AirPods - Apple Support
2. Scroll to: AirPods Pro 2 with MagSafe Charging Case (USB-C)
3. As it Reads: Model number: A3047, A3048, A3049

So, see if you have one of those models:

You can find the serial number of your AirPods in the same place as the model number. Follow the steps at the beginning of this article. If necessary, you can learn other ways to find the serial number.