let D1, D2 be LModMorphism_DOMAIN of G,H; :: thesis: ( ( for x being object holds
( x in D1 iff x is strict Morphism of G,H ) ) & ( for x being object holds
( x in D2 iff x is strict Morphism of G,H ) ) implies D1 = D2 )
assume that
A5: for x being object holds
( x in D1 iff x is strict Morphism of G,H ) and
A6: for x being object holds
( x in D2 iff x is strict Morphism of G,H ) ; :: thesis: D1 = D2
for x being object holds
( x in D1 iff x in D2 )
proof
let x be object ; :: thesis: ( x in D1 iff x in D2 )
hereby :: thesis: ( x in D2 implies x in D1 )
assume x in D1 ; :: thesis: x in D2
then x is strict Morphism of G,H by A5;
hence x in D2 by A6; :: thesis: verum
end;
assume x in D2 ; :: thesis: x in D1
then x is strict Morphism of G,H by A6;
hence x in D1 by A5; :: thesis: verum
end;
hence D1 = D2 by TARSKI:2; :: thesis: verum