consider f being Function such that
W1: dom f = R and
W2: for x being set st x in R holds
f . x = x `1 from FUNCT_1:sch 3();

now

let x be set ; :: thesis: ( ( x in rng f implies ex y being set st [x,y] in R ) & ( ex y being set st [x,y] in R implies x in rng f ) )
thus ( x in rng f implies ex y being set st [x,y] in R ) :: thesis: ( ex y being set st [x,y] in R implies x in rng f )

proof

assume x in rng f ; :: thesis: ex y being set st [x,y] in R
then consider a being set such that
W3: a in dom f and
W4: f . a = x by FUNCT_1:def 5;
take a `2 ; :: thesis: [x,(a `2 )] in R
F: ex x, y being set st a = [x,y] by W1, W3, RELAT_1:def 1;
x = a `1 by W2, W3, W4, W1;
hence [x,(a `2 )] in R by W1, W3, F, MCART_1:8; :: thesis: verum

end;

given y being set such that G: [x,y] in R ; :: thesis: x in rng f
f . [x,y] = [x,y] `1 by W2, G
.= x by MCART_1:7 ;
hence x in rng f by W1, G, FUNCT_1:12; :: thesis: verum

end;

then rng f = dom R by RELAT_1:def 4;
hence dom R is finite by W1, Th26; :: thesis: verum