let x be set ; :: thesis: for R1, R2 being Relation holds Im (R1 /\ R2),x = (Im R1,x) /\ (Im R2,x)
let R1, R2 be Relation; :: thesis: Im (R1 /\ R2),x = (Im R1,x) /\ (Im R2,x)
thus Im (R1 /\ R2),x c= (Im R1,x) /\ (Im R2,x) :: according to XBOOLE_0:def 10 :: thesis: (Im R1,x) /\ (Im R2,x) c= Im (R1 /\ R2),x

proof

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in Im (R1 /\ R2),x or y in (Im R1,x) /\ (Im R2,x) )
assume y in Im (R1 /\ R2),x ; :: thesis: y in (Im R1,x) /\ (Im R2,x)
then A1: [x,y] in R1 /\ R2 by Th9;
then A2: [x,y] in R1 by XBOOLE_0:def 4;
A3: [x,y] in R2 by A1, XBOOLE_0:def 4;
A4: y in Im R1,x by A2, Th9;
y in Im R2,x by A3, Th9;
hence y in (Im R1,x) /\ (Im R2,x) by A4, XBOOLE_0:def 4; :: thesis: verum

end;

let y be set ; :: according to TARSKI:def 3 :: thesis: ( not y in (Im R1,x) /\ (Im R2,x) or y in Im (R1 /\ R2),x )
assume A5: y in (Im R1,x) /\ (Im R2,x) ; :: thesis: y in Im (R1 /\ R2),x
then A6: y in Im R1,x by XBOOLE_0:def 4;
A7: y in Im R2,x by A5, XBOOLE_0:def 4;
A8: [x,y] in R1 by A6, Th9;
[x,y] in R2 by A7, Th9;
then [x,y] in R1 /\ R2 by A8, XBOOLE_0:def 4;
hence y in Im (R1 /\ R2),x by Th9; :: thesis: verum