
theorem
  for R being Relation, X,Y being set holds Y|`R|X = Y|`R /\ R|X
proof
  let R be Relation, X,Y be set;
  now
    let z be object;
    hereby
      assume A1: z in Y|`R|X;
      then consider x,y being object such that
        A2: z = [x,y] by RELAT_1:def 1;
      A3: z in Y|`R & x in X by A1, A2, RELAT_1:def 11;
      then z in R by A2, RELAT_1:def 12;
      then z in R|X by A2, A3, RELAT_1:def 11;
      hence z in Y|`R /\ R|X by A3, XBOOLE_0:def 4;
    end;
    assume z in Y|`R /\ R|X;
    then A4: z in Y|`R & z in R|X by XBOOLE_0:def 4;
    then consider x,y being object such that
      A5: z = [x,y] by RELAT_1:def 1;
    x in X by A4, A5, RELAT_1:def 11;
    hence z in Y|`R|X by A4, A5, RELAT_1:def 11;
  end;
  hence thesis by TARSKI:2;
end;
