
theorem Th39:
  for A,B being transitive with_units non empty AltCatStr,
  F being feasible FunctorStr over A,B st F is bijective Contravariant
  holds F" is Contravariant
proof
  let A,B be transitive with_units non empty AltCatStr,
  F be feasible FunctorStr over A,B;
  assume
A1: F is bijective Contravariant;
  then F is injective;
  then F is one-to-one;
  then
A2: the ObjectMap of F is one-to-one;
  F is surjective by A1;
  then F is onto;
  then
A3: the ObjectMap of F is onto;
  the ObjectMap of F is Contravariant by A1;
  then consider f being Function of the carrier of A, the carrier of B
  such that
A4: the ObjectMap of F = ~[:f,f:];
  [:f,f:] is one-to-one by A2,A4,Th9;
  then
A5: f is one-to-one by Th7;
  then
A6: dom(f") = rng f by FUNCT_1:33;
A7: rng(f") = dom f by A5,FUNCT_1:33;
  [:f,f:] is onto by A3,A4,Th13;
  then
A8: rng[:f,f:] = [:the carrier of B,the carrier of B:];
  rng[:f,f:] = [:rng f,rng f:] by FUNCT_3:67;
  then rng f = the carrier of B by A8,ZFMISC_1:92;
  then reconsider g = f" as Function of the carrier of B, the carrier of A
  by A6,A7,FUNCT_2:def 1,RELSET_1:4;
  take g;
A9: [:f,f:]" = [:g,g:] by A5,Th6;
  thus the ObjectMap of F" = (the ObjectMap of F)" by A1,Def38
    .= ~[:g,g:] by A4,A5,A9,Th10;
end;
