let FT be non empty RelStr ; :: thesis: for x being Element of FT
for A being Subset of FT holds
( x in A ^f iff ex y being Element of FT st
( P_A (y,A) = TRUE & P_0 (y,x) = TRUE ) )
let x be Element of FT; :: thesis: for A being Subset of FT holds
( x in A ^f iff ex y being Element of FT st
( P_A (y,A) = TRUE & P_0 (y,x) = TRUE ) )
let A be Subset of FT; :: thesis: ( x in A ^f iff ex y being Element of FT st
( P_A (y,A) = TRUE & P_0 (y,x) = TRUE ) )
A1: ( ex y being Element of FT st
( P_A (y,A) = TRUE & P_0 (y,x) = TRUE ) implies x in A ^f )

proof

assume ex y being Element of FT st
( P_A (y,A) = TRUE & P_0 (y,x) = TRUE ) ; :: thesis: x in A ^f
then consider y being Element of FT such that
A2: ( P_A (y,A) = TRUE & P_0 (y,x) = TRUE ) ;
( y in A & x in U_FT y ) by A2, Def3, Def4;
hence x in A ^f ; :: thesis: verum

end;

( x in A ^f implies ex y being Element of FT st
( P_A (y,A) = TRUE & P_0 (y,x) = TRUE ) )

proof

assume x in A ^f ; :: thesis: ex y being Element of FT st
( P_A (y,A) = TRUE & P_0 (y,x) = TRUE )
then consider y being Element of FT such that
A3: ( y in A & x in U_FT y ) by FIN_TOPO:11;
( P_A (y,A) = TRUE & P_0 (y,x) = TRUE ) by A3, Def3, Def4;
hence ex y being Element of FT st
( P_A (y,A) = TRUE & P_0 (y,x) = TRUE ) ; :: thesis: verum

end;

hence ( x in A ^f iff ex y being Element of FT st
( P_A (y,A) = TRUE & P_0 (y,x) = TRUE ) ) by A1; :: thesis: verum