Help finding smallest value (Personal bests for runners)

Hello,

I have a problem finding smallest value.

Practically this sheets is about athletes' running times. There's 16 people running, and I'd want this sheet to find their personal bests. Results (photo 1) will automatically go to an all-time list, which has all the times ever ran (photo 2).

As you can see in the photos, person N has ran two times. I tested using Vlookup (cell F5 in photo 1), but it only gives me N's first time, not their fastest time. (=VLOOKUP(C5,'All time results'!$W$4:$Y$300,3,FALSE())

Any ideas? I also tried index & match, but as it is not a function combo I've used that often, I only got out the fastest time ever ran, not the PBs of each runner.

This sheet is originally made in Google Sheets, but I made a simplified example file in Excel.

Thanks for help!