let Y be non empty set ; :: thesis: for a being Element of Funcs Y,BOOLEAN holds a '&' (I_el Y) = a
let a be Element of Funcs Y,BOOLEAN ; :: thesis: a '&' (I_el Y) = a
consider k3 being Function such that
A1: a '&' (I_el Y) = 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 (a '&' (I_el Y)) . x = a . x

proof

let x be Element of Y; :: thesis: (a '&' (I_el Y)) . x = a . x
(a . x) '&' ((I_el Y) . x) = (a . x) '&' TRUE by Def14;
hence (a '&' (I_el Y)) . x = a . x by MARGREL1:def 21; :: thesis: verum

end;

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