let Y be non empty set ; :: thesis: for a, b being Element of Funcs Y,BOOLEAN
for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y holds All (a 'eqv' b),PA,G = (All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)
let a, b be Element of Funcs Y,BOOLEAN ; :: thesis: for G being Subset of (PARTITIONS Y)
for PA being a_partition of Y holds All (a 'eqv' b),PA,G = (All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)
let G be Subset of (PARTITIONS Y); :: thesis: for PA being a_partition of Y holds All (a 'eqv' b),PA,G = (All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)
let PA be a_partition of Y; :: thesis: All (a 'eqv' b),PA,G = (All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)
A1:
for z being Element of Y holds (All (a 'eqv' b),PA,G) . z = ((All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)) . z
proof
let z be
Element of
Y;
:: thesis: (All (a 'eqv' b),PA,G) . z = ((All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)) . z
(All (a 'eqv' b),PA,G) . z =
(All ((a 'imp' b) '&' (b 'imp' a)),PA,G) . z
by Th7
.=
((All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)) . z
by BVFUNC_1:42
;
hence
(All (a 'eqv' b),PA,G) . z = ((All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)) . z
;
:: thesis: verum
end;
consider k3 being Function such that
A2:
( All (a 'eqv' b),PA,G = k3 & dom k3 = Y & rng k3 c= BOOLEAN )
by FUNCT_2:def 2;
consider k4 being Function such that
A3:
( (All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G) = k4 & dom k4 = Y & rng k4 c= BOOLEAN )
by FUNCT_2:def 2;
( Y = dom k3 & Y = dom k4 & ( for u being set st u in Y holds
k3 . u = k4 . u ) )
by A1, A2, A3;
hence
All (a 'eqv' b),PA,G = (All (a 'imp' b),PA,G) '&' (All (b 'imp' a),PA,G)
by A2, A3, FUNCT_1:9; :: thesis: verum