let Y be non empty set ; :: thesis: for a being Element of Funcs (Y,BOOLEAN) holds (I_el Y) 'imp' a = a
let a be Element of Funcs (Y,BOOLEAN); :: thesis: (I_el Y) 'imp' a = a
consider k3 being Function such that
A1: (I_el Y) 'imp' a = k3 and
A2: dom k3 = Y and
rng k3 c= BOOLEAN by FUNCT_2:def 2;
consider k4 being Function such that
A3: a = k4 and
A4: dom k4 = Y and
rng k4 c= BOOLEAN by FUNCT_2:def 2;
for x being Element of Y holds ((I_el Y) 'imp' a) . x = a . x

proof

let x be Element of Y; :: thesis: ((I_el Y) 'imp' a) . x = a . x
((I_el Y) 'imp' a) . x = ('not' ((I_el Y) . x)) 'or' (a . x) by BVFUNC_1:def 11
.= ('not' TRUE) 'or' (a . x) by BVFUNC_1:def 14
.= FALSE 'or' (a . x) by MARGREL1:41 ;
hence ((I_el Y) 'imp' a) . x = a . x by BINARITH:7; :: thesis: verum

end;

then for u being set st u in Y holds
k3 . u = k4 . u by A1, A3;
hence (I_el Y) 'imp' a = a by A1, A2, A3, A4, FUNCT_1:9; :: thesis: verum