reserve Y,Z for non empty set;
reserve PA,PB for a_partition of Y;
reserve A,B for Subset of Y;
reserve i,j,k for Nat;
reserve x,y,z,x1,x2,y1,z0,X,V,a,b,d,t,SFX,SFY for set;

theorem
  %I(Y)={B:ex x being set st B={x} & x in Y}
proof
  set B0={B:ex x being set st B={x} & x in Y};
A1: %I(Y) c= B0
  proof
    let a be object;
    assume a in %I(Y); then
    a in the set of all {x} where x is Element of Y by EQREL_1:37;
    then consider x be Element of Y such that
A2: a={x};
    reconsider y=x as Element of Y;
    reconsider B={y} as Subset of Y by ZFMISC_1:31;
    a=B by A2;
    hence thesis;
  end;
  B0 c= %I(Y)
  proof
    let x1 be object;
    assume x1 in B0;
    then ex B st x1=B & ex x being set st B={x} & x in Y;
    then x1 in the set of all {z} where z is Element of Y;
    hence thesis by EQREL_1:37;
  end;
  hence thesis by A1,XBOOLE_0:def 10;
end;
