Question about Kaffarah (Hanafi Madhab)

[a.s.]

Someone tell me if this is right.
Kaffarah has to be made when a ramadan fast is broken intentionally, whether through sexual intercourse or by any other means.
So if you eat something during a ramadan fast, knowing that you are fasting, then you have to do the kaffarah?

So kaffarah is necessary if all of the following is true:
1) The fast was broken intentionally, in whatever way.
2) The fast was during Ramadan.
3) Intention for the fast was made before Fajr the previous night, and NOT after Fajr.

is this correct?

Hanafi Fiqh Only

[a.s.]

anyone?