reserve A,X,X1,X2,Y,Y1,Y2 for set, a,b,c,d,x,y,z for object;
reserve P,P1,P2,Q,R,S for Relation;

theorem Th78:
  y in rng(Y|`R) iff y in Y & y in rng R
proof
  thus y in rng(Y|`R) implies y in Y & y in rng R
  proof
    assume y in rng(Y|`R);
    then consider x being object such that
A1: [x,y] in Y|`R by XTUPLE_0:def 13;
    [x,y] in R by A1,Def10;
    hence thesis by A1,Def10,XTUPLE_0:def 13;
  end;
  assume
A2: y in Y;
  assume y in rng R;
  then consider x being object such that
A3: [x,y] in R by XTUPLE_0:def 13;
  [x,y] in Y|`R by A2,A3,Def10;
  hence thesis by XTUPLE_0:def 13;
end;
