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;

theorem Th14:
  x in B implies (A --> x)"B = A
proof
  assume
A1: x in B;
  now
    per cases;
    suppose A = {};
      hence thesis;
    end;
    suppose
      A <> {};
      then
A3:   rng (A --> x) = {x} by RELAT_1:160;
      {x} c= B by A1,ZFMISC_1:31;
      then {x} /\ B = {x} by XBOOLE_1:28;
      hence (A --> x)"B = (A --> x)"{x} by A3,RELAT_1:133
        .= dom (A -->x) by A3,RELAT_1:134
        .= A;
    end;
  end;
  hence thesis;
end;
