Is a brute force method considered a proof? Say we have some finite set, and some theory about a set, say "All elements of the finite set $X$ satisfy condition $Y$".
If we let a computer check every single member of $X$ and conclude that the condition $Y$ holds for all of them, can we call this a proof? Or is it possibly something else?

Yes, you can. This method is known as proof by exhaustion.
Also, see computer-assisted proof.
Edit: As others have noted, this of course works only for finite sets.
