reserve G for _Graph;

theorem
  for v being Vertex of G holds Im(VertexAdjSymRel(G),v) = v.allNeighbors()
proof
  let v be Vertex of G;
  now
    let x be object;
    hereby
      assume x in (VertexAdjSymRel(G)).:{v};
      then consider v0 being object such that
        A1: [v0,x] in VertexAdjSymRel(G) & v0 in {v} by RELAT_1:def 13;
      [v,x] in VertexAdjSymRel(G) by A1, TARSKI:def 1;
      then x is set & ex e being object st e Joins v,x,G by Th32, TARSKI:1;
      hence x in v.allNeighbors() by GLIB_000:71;
    end;
    assume x in v.allNeighbors();
    then ex e being object st e Joins v,x,G by GLIB_000:71;
    then [v,x] in VertexAdjSymRel(G) & v in {v} by Th32, TARSKI:def 1;
    hence x in (VertexAdjSymRel(G)).:{v} by RELAT_1:def 13;
  end;
  then (VertexAdjSymRel(G)).:{v} = v.allNeighbors() by TARSKI:2;
  hence thesis by RELAT_1:def 16;
end;
