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
( [x,y] in EqR 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
( [x,y] in EqR iff Class (EqR,x) = Class (EqR,y) )
let EqR be Equivalence_Relation of X; :: thesis: for x being object st x in X holds
( [x,y] in EqR iff Class (EqR,x) = Class (EqR,y) )
let x be object ; :: thesis: ( x in X implies ( [x,y] in EqR iff Class (EqR,x) = Class (EqR,y) ) )
assume A1: x in X ; :: thesis: ( [x,y] in EqR iff Class (EqR,x) = Class (EqR,y) )
thus ( [x,y] in EqR implies Class (EqR,x) = Class (EqR,y) ) :: thesis: ( Class (EqR,x) = Class (EqR,y) implies [x,y] in EqR )
proof
assume A2: [x,y] in EqR ; :: thesis: Class (EqR,x) = Class (EqR,y)
now :: thesis: for z being object st z in Class (EqR,y) holds
z in Class (EqR,x)
let z be object ; :: thesis: ( z in Class (EqR,y) implies z in Class (EqR,x) )
assume z in Class (EqR,y) ; :: thesis: z in Class (EqR,x)
then A3: [z,y] in EqR by Th19;
[y,x] in EqR by A2, Th6;
then [z,x] in EqR by A3, Th7;
hence z in Class (EqR,x) by Th19; :: thesis: verum
end;
then A4: Class (EqR,y) c= Class (EqR,x) ;
now :: thesis: for z being object st z in Class (EqR,x) holds
z in Class (EqR,y)
let z be object ; :: thesis: ( z in Class (EqR,x) implies z in Class (EqR,y) )
assume z in Class (EqR,x) ; :: thesis: z in Class (EqR,y)
then [z,x] in EqR by Th19;
then [z,y] in EqR by A2, Th7;
hence z in Class (EqR,y) by Th19; :: thesis: verum
end;
then Class (EqR,x) c= Class (EqR,y) ;
hence Class (EqR,x) = Class (EqR,y) by A4; :: thesis: verum
end;
assume Class (EqR,x) = Class (EqR,y) ; :: thesis: [x,y] in EqR
then x in Class (EqR,y) by A1, Th20;
hence [x,y] in EqR by Th19; :: thesis: verum