
theorem NF501:
  for x, y being object holds {[x, y]} " {y} = {x}
  proof
    let x, y be object;

    for v being object holds
    v in {x} iff (v in dom {[x, y]} & {[x, y]} . v in {y})
    proof
      let v be object;

      hereby
        assume v in {x};
        then L032: v = x by TARSKI:def 1;

        L0325: [x, y] in {[x, y]} by TARSKI:def 1;
        then L033: x in dom {[x, y]} & {[x, y]} . x = y by FUNCT_1:1;

        thus v in dom {[x, y]} by L032,L0325,FUNCT_1:1;
        thus {[x, y]} . v in {y} by L033,TARSKI:def 1,L032;
      end;

      assume v in dom {[x, y]} & {[x, y]} . v in {y};
      hence v in {x} by RELAT_1:9;
    end;
    hence thesis by FUNCT_1:def 7;
  end;
