reserve x,y,y1,y2,z,a,b for object, X,Y,Z,V1,V2 for set,
  f,g,h,h9,f1,f2 for Function,
  i for Nat,
  P for Permutation of X,
  D,D1,D2,D3 for non empty set,
  d1 for Element of D1,
  d2 for Element of D2,
  d3 for Element of D3;

theorem Th20:
  doms (X --> f) = X --> dom f & rngs (X --> f) = X --> rng f
proof
A1: dom (X --> dom f) = X &
  dom doms (X --> f) = dom (X --> f ) by Def1;
A2: dom (X --> f) = X & (X --> f)"rng (X --> f) = dom (X --> f) by RELAT_1:134;
  now
    let x be object;
    assume
A3: x in X;
    then (X --> f).x = f & (X --> dom f).x = dom f by FUNCOP_1:7;
    hence (doms (X --> f)).x = (X --> dom f).x by A2,A3,Def1;
  end;
  hence doms (X --> f) = X --> dom f by A1;
A4: now
    let x be object;
    assume
A5: x in X;
    then (X --> f).x = f & (X --> rng f).x = rng f by FUNCOP_1:7;
    hence (rngs (X --> f)).x = (X --> rng f).x by A2,A5,Def2;
  end;
  dom (X --> rng f) = X &
   dom rngs (X --> f) = dom (X --> f ) by Def2;
  hence thesis by A4;
end;
