reserve a,b,p,x,x9,x1,x19,x2,y,y9,y1,y19,y2,z,z9,z1,z2 for object,
   X,X9,Y,Y9,Z,Z9 for set;
reserve A,D,D9 for non empty set;
reserve f,g,h for Function;

theorem Th52:
  dom f c= [:X,Y:] implies ~~f = f
proof
  assume
A1: dom f c= [:X,Y:];
A2: ~~f c= f by Th51;
  dom ~~ f = dom f
  proof
    thus dom ~~f c= dom f by A2,RELAT_1:11;
    let z be object;
    assume
A3: z in dom f;
    then consider x,y being object such that
    x in X and
    y in Y and
A4: z = [x,y] by A1,ZFMISC_1:84;
    [y,x] in dom ~f by A3,A4,Th42;
    hence thesis by A4,Th42;
  end;
  hence thesis by Th51,GRFUNC_1:3;
end;
