How would you only be paying $6k/year property tax? I'd expect more than $10k/year or around 850/month.
The property tax rate is 1.1826% of assessed value in SF (1% state + a small % local). And your initial assessed value is based on your purchase price.
I think it's more realistic to say you'd be spending close to 4500/month net assuming you didn't live in a place with an HOA. If you live in a condo with an HOA, you could easily be spending 5000 to 5500/month.
On the plus side, at least you get to deduct all that mortgage interest and property tax on your federal taxes. When you are paying 3k/month+ in interest and taxes, that's a really nice tax advantage over renting a similar property.