let F1, F2 be FUNCTION_DOMAIN of the carrier of UA, the carrier of UA; :: thesis: ( ( for h being Function of UA,UA holds

( h in F1 iff h is_isomorphism ) ) & ( for h being Function of UA,UA holds

( h in F2 iff h is_isomorphism ) ) implies F1 = F2 )

assume that

A3: for h being Function of UA,UA holds

( h in F1 iff h is_isomorphism ) and

A4: for h being Function of UA,UA holds

( h in F2 iff h is_isomorphism ) ; :: thesis: F1 = F2

A5: F2 c= F1

( h in F1 iff h is_isomorphism ) ) & ( for h being Function of UA,UA holds

( h in F2 iff h is_isomorphism ) ) implies F1 = F2 )

assume that

A3: for h being Function of UA,UA holds

( h in F1 iff h is_isomorphism ) and

A4: for h being Function of UA,UA holds

( h in F2 iff h is_isomorphism ) ; :: thesis: F1 = F2

A5: F2 c= F1

proof

F1 c= F2
let q be object ; :: according to TARSKI:def 3 :: thesis: ( not q in F2 or q in F1 )

assume A6: q in F2 ; :: thesis: q in F1

then reconsider h1 = q as Function of UA,UA by FUNCT_2:def 12;

h1 is_isomorphism by A4, A6;

hence q in F1 by A3; :: thesis: verum

end;assume A6: q in F2 ; :: thesis: q in F1

then reconsider h1 = q as Function of UA,UA by FUNCT_2:def 12;

h1 is_isomorphism by A4, A6;

hence q in F1 by A3; :: thesis: verum

proof

hence
F1 = F2
by A5; :: thesis: verum
let q be object ; :: according to TARSKI:def 3 :: thesis: ( not q in F1 or q in F2 )

assume A7: q in F1 ; :: thesis: q in F2

then reconsider h1 = q as Function of UA,UA by FUNCT_2:def 12;

h1 is_isomorphism by A3, A7;

hence q in F2 by A4; :: thesis: verum

end;assume A7: q in F1 ; :: thesis: q in F2

then reconsider h1 = q as Function of UA,UA by FUNCT_2:def 12;

h1 is_isomorphism by A3, A7;

hence q in F2 by A4; :: thesis: verum