
theorem Th22:
  for R,S being non empty doubleLoopStr, I being Ideal of R, P
  being Function of R,S st P is RingIsomorphism holds P.:I is Ideal of S
proof
  let R,S being non empty doubleLoopStr, I being Ideal of R, P being Function
  of R,S;
  set Q = P.:I;
  assume
A1: P is RingIsomorphism;
A2: now
    let p, x be Element of S;
    assume x in Q;
    then consider x9 being object such that
A3: x9 in the carrier of R and
A4: x9 in I and
A5: x = P.x9 by FUNCT_2:64;
    reconsider x9 as Element of R by A3;
A6: P is RingMonomorphism RingEpimorphism by A1;
    then P is onto;
    then consider p9 being object such that
A7: p9 in dom P and
A8: p = P.p9 by FUNCT_1:def 3;
    reconsider p9 as Element of R by A7;
A9: p9*x9 in I by A4,IDEAL_1:def 2;
    P is RingHomomorphism by A6;
    then P is multiplicative;
    then p*x = P.(p9*x9) by A5,A8;
    hence p*x in Q by A9,FUNCT_2:35;
  end;
A10: now
    let p, x be Element of S;
    assume x in Q;
    then consider x9 being object such that
A11: x9 in the carrier of R and
A12: x9 in I and
A13: x = P.x9 by FUNCT_2:64;
    reconsider x9 as Element of R by A11;
A14: P is RingMonomorphism RingEpimorphism by A1;
    then P is onto;
    then consider p9 being object such that
A15: p9 in dom P and
A16: p = P.p9 by FUNCT_1:def 3;
    reconsider p9 as Element of R by A15;
A17: x9*p9 in I by A12,IDEAL_1:def 3;
    P is RingHomomorphism by A14;
    then P is multiplicative;
    then x*p = P.(x9*p9) by A13,A16;
    hence x*p in Q by A17,FUNCT_2:35;
  end;
  now
    let x, y be Element of S;
    assume that
A18: x in Q and
A19: y in Q;
    consider x9 being object such that
A20: x9 in the carrier of R and
A21: x9 in I and
A22: x = P.x9 by A18,FUNCT_2:64;
    consider y9 being object such that
A23: y9 in the carrier of R and
A24: y9 in I and
A25: y = P.y9 by A19,FUNCT_2:64;
    reconsider x9,y9 as Element of R by A20,A23;
A26: x9+y9 in I by A21,A24,IDEAL_1:def 1;
    P is RingMonomorphism RingEpimorphism by A1;
    then P is additive;
    then x+y = P.(x9+y9) by A22,A25;
    hence x+y in Q by A26,FUNCT_2:35;
  end;
  hence thesis by A2,A10,IDEAL_1:def 1,def 2,def 3;
end;
