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

a 'imp' b = I_el Y

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

assume A1: b = I_el Y ; :: thesis: a 'imp' b = I_el Y

for x being Element of Y holds (a 'imp' b) . x = TRUE

a 'imp' b = I_el Y

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

assume A1: b = I_el Y ; :: thesis: a 'imp' b = I_el Y

for x being Element of Y holds (a 'imp' b) . x = TRUE

proof

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

b . x = TRUE by A1, BVFUNC_1:def 11;

then (a 'imp' b) . x = ('not' (a . x)) 'or' TRUE by BVFUNC_1:def 8

.= TRUE by BINARITH:10 ;

hence (a 'imp' b) . x = TRUE ; :: thesis: verum

end;b . x = TRUE by A1, BVFUNC_1:def 11;

then (a 'imp' b) . x = ('not' (a . x)) 'or' TRUE by BVFUNC_1:def 8

.= TRUE by BINARITH:10 ;

hence (a 'imp' b) . x = TRUE ; :: thesis: verum