reserve X,Y,Z,X1,X2,Y1,Y2 for set, x,y,z,t,x1,x2 for object,
  f,g,h,f1,f2,g1,g2 for Function;

theorem Th30:
  rng f c= PFuncs(X,Y) implies dom uncurry f c= [:dom f,X:] & dom
  uncurry' f c= [:X,dom f:]
proof
  assume
A1: rng f c= PFuncs(X,Y);
  thus
A2: dom uncurry f c= [:dom f,X:]
  proof
    let x be object;
    assume x in dom uncurry f;
    then consider y,g,z such that
A3: x = [y,z] and
A4: y in dom f and
A5: g = f.y and
A6: z in dom g by Def2;
    g in rng f by A4,A5,FUNCT_1:def 3;
    then g is PartFunc of X,Y by A1,PARTFUN1:46;
    then dom g c= X by RELAT_1:def 18;
    hence thesis by A3,A4,A6,ZFMISC_1:87;
  end;
  let x be object;
  assume x in dom uncurry' f;
  then ex y,z being object st x = [z,y] & [y,z] in dom uncurry f
    by FUNCT_4:def 2;
  hence thesis by A2,ZFMISC_1:88;
end;
