Unless it is an offically licenced product by the manufacturer; then you may have problems. I truly wish you the best of luck. And as an aside, if you found my direction helpful, and if you feel appropriate; could you be so kind as to designate my answer as the “best” answer to your question?
I don't know whether the BIOS is copyright protected, so I can't really say. You say the "Internet," but I have a feeling you mean from a bittorrent client. So it depends on the "source." If you are using bittorents, you are gambling because you may get sued. If you get sued it could cost you a lot more than a new Play Station. I say this not just for the downloading of the bios, but also for games, movies and music. I know because one of my practice areas is BitTorrent defense, and I have seen thousands of people get sued over allegedly unauthorized downloads. A lot of times people don't just use Torrents for one thing; the more they download, the easier it becomes, so they download more, then they get caught. So I always advise against it.
There are ways to download or stream licensed games from the Internet that are not infringing copyrights. For example, Steam has some free options as well as low-cost ones. I'm not a big gamer, but I have used it myself for a couple of games, and it's not bad. Same with movies - plenty of legal options out there.
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NO. That's not likely legal. The Basic Input/Output Software (BIOS) for a PlayStation 2 is likely not open source but rather copyright protected. For those attorneys not familiar with computer technology, you can read how BIOS works here: http://www.howstuffworks.com/bios.htm
The fact that you threw away a proprietary piece of software does not give you the right to steal a copy via BitTorrent (dangerous, as copyright trolls are tracking much of it) or from some unauthorized source via Dropbox or other cloud service.
While not legal, it's highly unlikely you would get busted for this, although I am not sure why you would need the BIOS if you don't have the actual PS2. Seems to me the emulator would use the platform BIOS of the platform on which the emulator runs. Sounds like a lousy emulator if you ask me.
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