let x, y be object ; :: thesis: for R being Relation holds

( y in Class (R,x) iff [x,y] in R )

let R be Relation; :: thesis: ( y in Class (R,x) iff [x,y] in R )

thus ( y in Class (R,x) implies [x,y] in R ) :: thesis: ( [x,y] in R implies y in Class (R,x) )

assume [x,y] in R ; :: thesis: y in Class (R,x)

hence y in Class (R,x) by A1, RELAT_1:def 13; :: thesis: verum

( y in Class (R,x) iff [x,y] in R )

let R be Relation; :: thesis: ( y in Class (R,x) iff [x,y] in R )

thus ( y in Class (R,x) implies [x,y] in R ) :: thesis: ( [x,y] in R implies y in Class (R,x) )

proof

A1:
x in {x}
by TARSKI:def 1;
assume
y in Class (R,x)
; :: thesis: [x,y] in R

then ex z being object st

( [z,y] in R & z in {x} ) by RELAT_1:def 13;

hence [x,y] in R by TARSKI:def 1; :: thesis: verum

end;then ex z being object st

( [z,y] in R & z in {x} ) by RELAT_1:def 13;

hence [x,y] in R by TARSKI:def 1; :: thesis: verum

assume [x,y] in R ; :: thesis: y in Class (R,x)

hence y in Class (R,x) by A1, RELAT_1:def 13; :: thesis: verum