let Y be non empty set ; :: thesis: for a being Element of Funcs (Y,BOOLEAN) holds (O_el Y) 'imp' a = I_el Y
let a be Element of Funcs (Y,BOOLEAN); :: thesis: (O_el Y) 'imp' a = I_el Y
for x being Element of Y holds ((O_el Y) 'imp' a) . x = TRUE

proof

let x be Element of Y; :: thesis: ((O_el Y) 'imp' a) . x = TRUE
((O_el Y) 'imp' a) . x = ('not' ((O_el Y) . x)) 'or' (a . x) by BVFUNC_1:def 11
.= ('not' FALSE) 'or' (a . x) by BVFUNC_1:def 13
.= TRUE 'or' (a . x) by MARGREL1:41 ;
hence ((O_el Y) 'imp' a) . x = TRUE by BINARITH:19; :: thesis: verum

end;

hence (O_el Y) 'imp' a = I_el Y by BVFUNC_1:def 14; :: thesis: verum