let F1, F2 be set ; :: thesis: ( ( for x being set holds
( x in F1 iff ex f being Function st
( x = f & dom f c= X & rng f c= Y ) ) ) & ( for x being set holds
( x in F2 iff ex f being Function st
( x = f & dom f c= X & rng f c= Y ) ) ) implies F1 = F2 )
assume that
A6: for x being set holds
( x in F1 iff ex f being Function st
( x = f & dom f c= X & rng f c= Y ) ) and
A7: for x being set holds
( x in F2 iff ex f being Function st
( x = f & dom f c= X & rng f c= Y ) ) ; :: thesis: F1 = F2
for x being set holds
( x in F1 iff x in F2 )

proof

let x be set ; :: thesis: ( x in F1 iff x in F2 )
( x in F1 iff ex f being Function st
( x = f & dom f c= X & rng f c= Y ) ) by A6;
hence ( x in F1 iff x in F2 ) by A7; :: thesis: verum

end;

hence F1 = F2 by TARSKI:1; :: thesis: verum