let X be set ; :: thesis: for y being object
for EqR being Equivalence_Relation of X
for x being object st x in X holds
( y in Class (EqR,x) iff Class (EqR,x) = Class (EqR,y) )
let y be object ; :: thesis: for EqR being Equivalence_Relation of X
for x being object st x in X holds
( y in Class (EqR,x) iff Class (EqR,x) = Class (EqR,y) )
let EqR be Equivalence_Relation of X; :: thesis: for x being object st x in X holds
( y in Class (EqR,x) iff Class (EqR,x) = Class (EqR,y) )
let x be object ; :: thesis: ( x in X implies ( y in Class (EqR,x) iff Class (EqR,x) = Class (EqR,y) ) )
assume A1: x in X ; :: thesis: ( y in Class (EqR,x) iff Class (EqR,x) = Class (EqR,y) )
thus ( y in Class (EqR,x) implies Class (EqR,x) = Class (EqR,y) ) :: thesis: ( Class (EqR,x) = Class (EqR,y) implies y in Class (EqR,x) )
proof
assume y in Class (EqR,x) ; :: thesis: Class (EqR,x) = Class (EqR,y)
then [y,x] in EqR by Th19;
then [x,y] in EqR by Th6;
hence Class (EqR,x) = Class (EqR,y) by A1, Lm2; :: thesis: verum
end;
assume Class (EqR,x) = Class (EqR,y) ; :: thesis: y in Class (EqR,x)
then [x,y] in EqR by A1, Lm2;
then [y,x] in EqR by Th6;
hence y in Class (EqR,x) by Th19; :: thesis: verum