Some people have talked about "just showing up and playing whatever is provided". I think that has more to do with the player than the gear, and I do think that is applicable in some situations, but not in others.
As a sound guy, when a band shows up and they have gear that is clearly deliberately chosen because it's the best thing available (let's say within a reasonable budget, on a sliding scale based on the level of the band/venue/etc...) for what the band is trying to do, that demonstrates a level of professionalism, and unless there were exigent circumstances, I would never force such a band to use the backline if they didn't want to. That doesn't mean the most expensive or the biggest or the most power, it means, well chosen for the specific application to the extent that there would be little if any benefit of using something else. So if your blues band shows up and the guitarist has a 120w Diezel tube full stack that is not "well chosen". Likewise, if your metal band shows up and the guitarist has a little Blues Jr, that is also likely not "well chosen" (in some settings maybe it could work).
So it's really about finding gear that nails what you are trying to do. The more closely it's nailed, the more pro it is essentially.
Obviously there is also the matter of power level (varies, but should be sufficient for the venue/style/etc... and be able to keep up easily but without being ridiculously over powered), reliability (this can also be hedged with backups for key items), etc... So really to some extent the definition of "pro gear" is that it does exactly what it needs to do, extremely well, and extremely reliably.
Getting back to the player, part of being a pro is being able to roll in with your own gear, see the provided backline, and make a wise and prudent determination of which you should use, and be right 90% of the time or better. That will vary by situation and circumstances. If you're a small local band playing a big festival stage with quick changeover times, you may have to "settle" for the backline. On the other hand, if it is really junky, you may have to insist on your own gear if you can make it work, but then you also need to make the changeover as seamless as possible. If you know the guitarist or keyboardist or whoever is going to take X time to get THEIR stuff ready, and you know you can still be ready in X-5, that's probably fine, just don't be the one holding up the proceedings.
Finally, for an idea of a "generic" version of an all around "pro" bass amp setup that should work in almost any settings, I would say it should have ideally at least 300-500w, and the equivalent of 2x12s or 4x10s. A good sounding preamp that can hopefully do somewhat neutral as well as any color you might want (hybrid heads are good for this), and have a flexible EQ, plus whatever key features you expect to need on a gig to gig basis (ie things like built in compression, or input pads, effects loop, etc...). Lastly, it should not be the cheapest crappiest thing you can find. You want to do extensive research and max out your budget on the best VALUE (not most expensive) pieces you can find. Used is almost ALWAYS better for this if you are on a budget. My whole "big rig", depending on which head I use cost me in the $1500-2k range. That's enough to play probably anything short of arenas (and maybe those too, considering there would PA support). But it is all VERY NICE and great sounding and flexible gear. I have smaller configurations down to the $500-1000 range that also sound great. Because I bought it all used.
And depending on your own playing situation/s, being a pro may also mean having a handful of different configurations available depending on the specific gig. If I had a low to medium wattage tube head added to what I have now, I would basically be ready for anything. And I actually used 2 of my heads and 2 of my cabs for reamping the bass for a band I'm now (but playing keys). And between those 4 possible combinations, we got a nice range of different sounds between the different tracks (and I had another much more different head I wanted to use too, but I didn't have it handy, as it's at a different location).