reserve f,g,h for Function,
  A for set;
reserve F for Function,
  B,x,y,y1,y2,z for set;
reserve x,z for object;
reserve X for non empty set,
  Y for set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;
reserve Y for non empty set,
  F for BinOp of X,
  f,g,h for Function of Y,X,
  x,x1,x2 for Element of X;

theorem Th45:
  x in B implies A --> x is Function of A,B
proof
A1: rng (A --> x) c= {x} by Th13;
A2: dom (A --> x) = A;
  assume
A3: x in B;
  then {x} c= B by ZFMISC_1:31;
  then rng (A --> x) c= B by A1;
  hence thesis by A3,A2,FUNCT_2:def 1,RELSET_1:4;
end;
