let Y be non empty set ; :: thesis: for a being Function of Y,BOOLEAN holds (O_el Y) 'imp' a = I_el Y
let a be Function of 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 8
.= TRUE 'or' (a . x) by BVFUNC_1:def 10 ;
hence ((O_el Y) 'imp' a) . x = TRUE ; :: thesis: verum

end;

hence (O_el Y) 'imp' a = I_el Y by BVFUNC_1:def 11; :: thesis: verum