New kind of puzzle!

Ok so I was thinking about a checkmate where every peice, I mean non-pawns, are essential. So if you remove one peice from the board it is no longer mate. First I was trying to find one without any pawns on the board but I couldn't. Once I allowed pawns it became easy to find. So, this is what I found:

 Am I even right? Is every piece nesassary? Can you do it without a pawn? 

EDIT: this is what it says beneath the board: Mate. If you remove the White king than the black king can go to c7, if you remove the a4bishop than the king can go to c6, if you remove the queen, than the king can go to e5, if you remove the Rook on d8 than the king can take on d7, if you remove the d7 pawn than the e3 knight would become unnecessary if you remove the e3 knight than the king can go to d5 if you remove the g5 knight than the king could go to e6 and if you remove the h7 rook than the king could take one7. If you remove the Bishop on e7 than its not mate.

Thank you:)