Die hard = Fanatical

執着する。熱狂的な。

Osaka is filled with die hard Tigers fans.

A feather in your cap = A reward of some kind

誇りとなるもの、手柄。

After a great deal of difficulty, the salesman got his order. That was a feather in his cap.

To bounce back = To recover

回復する。

After hard training, it's difficult for me to bounce back.

To be down = Not to operate

ダウンしている。

Since my P.C. is old, it's down more than it's up.

To be up for = To want something

〜するつもり。

After training I'm always up for a cold drink.

To be on the ball = To be ready or to be aware

注意が行き届いて、油断のない。

If you want to succeed in this world, you must be on the ball at all times.

To be whipped = To be very tired

疲労困憊して。

After a tough day, I am really whipped.

To rock on = To continue even though it's difficult

思う存分楽しむ。

If life is getting you down, take a break and then rock on.

To dim = To turn down light, power etc.

（・・・を）薄暗くする。

To save electricity, we must dim our lights and air conditioning.

To have your heart set on something = To want to do something very, very badly

切望する。

I have my heart on going to Chicago to watch my daughter dance.