let Y be non empty set ; :: thesis: for b being Function of Y,BOOLEAN st (I_el Y) 'imp' b = I_el Y holds

b = I_el Y

set a = I_el Y;

let b be Function of Y,BOOLEAN; :: thesis: ( (I_el Y) 'imp' b = I_el Y implies b = I_el Y )

assume A1: (I_el Y) 'imp' b = I_el Y ; :: thesis: b = I_el Y

for x being Element of Y holds b . x = TRUE

b = I_el Y

set a = I_el Y;

let b be Function of Y,BOOLEAN; :: thesis: ( (I_el Y) 'imp' b = I_el Y implies b = I_el Y )

assume A1: (I_el Y) 'imp' b = I_el Y ; :: thesis: b = I_el Y

for x being Element of Y holds b . x = TRUE

proof

hence
b = I_el Y
by BVFUNC_1:def 11; :: thesis: verum
let x be Element of Y; :: thesis: b . x = TRUE

((I_el Y) 'imp' b) . x = TRUE by A1, BVFUNC_1:def 11;

then ( (I_el Y) . x = TRUE & ('not' ((I_el Y) . x)) 'or' (b . x) = TRUE ) by BVFUNC_1:def 8, BVFUNC_1:def 11;

then FALSE 'or' (b . x) = TRUE by MARGREL1:11;

hence b . x = TRUE by BINARITH:3; :: thesis: verum

end;((I_el Y) 'imp' b) . x = TRUE by A1, BVFUNC_1:def 11;

then ( (I_el Y) . x = TRUE & ('not' ((I_el Y) . x)) 'or' (b . x) = TRUE ) by BVFUNC_1:def 8, BVFUNC_1:def 11;

then FALSE 'or' (b . x) = TRUE by MARGREL1:11;

hence b . x = TRUE by BINARITH:3; :: thesis: verum