reserve X,Y for set;
reserve Z for non empty set;

theorem Th5:
  for A,B being non empty AltCatStr, F being FunctorStr over A,B st
  F is bijective holds the ObjectMap of F is bijective & the MorphMap of F is
  "1-1"
proof
  let A,B be non empty AltCatStr, F be FunctorStr over A,B;
  assume
A1: F is bijective;
  then
A2: F is injective;
  then F is one-to-one;
  then
A3: the ObjectMap of F is one-to-one;
  F is surjective by A1;
  then F is onto;
  then
A4: (the ObjectMap of F) is onto;
  F is faithful by A2;
  hence thesis by A3,A4;
end;
