set S = { [F₃(x),F₄(x)] where x is Element of F₁() : P₁[x] } ;
{ [F₃(x),F₄(x)] where x is Element of F₁() : P₁[x] } is Relation-like

proof

let x be set ; :: according to RELAT_1:def 1 :: thesis: ( not x in { [F₃(x),F₄(x)] where x is Element of F₁() : P₁[x] } or ex b₁, b₂ being set st x = [b₁,b₂] )
assume x in { [F₃(x),F₄(x)] where x is Element of F₁() : P₁[x] } ; :: thesis: ex b₁, b₂ being set st x = [b₁,b₂]
then ex y being Element of F₁() st
( x = [F₃(y),F₄(y)] & P₁[y] ) ;
hence ex b₁, b₂ being set st x = [b₁,b₂] ; :: thesis: verum

end;

then reconsider S' = { [F₃(x),F₄(x)] where x is Element of F₁() : P₁[x] } as Relation ;
for x, y being set holds
( [x,y] in S' iff [y,x] in F₂() )

proof

A2: for x, y being set st [x,y] in S' holds
[y,x] in F₂()

proof

let x, y be set ; :: thesis: ( [x,y] in S' implies [y,x] in F₂() )
assume [x,y] in S' ; :: thesis: [y,x] in F₂()
then consider z being Element of F₁() such that
A3: [x,y] = [F₃(z),F₄(z)] and
A4: P₁[z] ;
( x = F₃(z) & y = F₄(z) ) by A3, ZFMISC_1:33;
hence [y,x] in F₂() by A1, A4; :: thesis: verum

end;

for x, y being set st [y,x] in F₂() holds
[x,y] in S'

proof

let x, y be set ; :: thesis: ( [y,x] in F₂() implies [x,y] in S' )
assume [y,x] in F₂() ; :: thesis: [x,y] in S'
then consider z being Element of F₁() such that
A5: [y,x] = [F₄(z),F₃(z)] and
A6: P₁[z] by A1;
( y = F₄(z) & x = F₃(z) ) by A5, ZFMISC_1:33;
hence [x,y] in S' by A6; :: thesis: verum

end;

hence for x, y being set holds
( [x,y] in S' iff [y,x] in F₂() ) by A2; :: thesis: verum

end;

hence F₂() ~ = { [F₃(x),F₄(x)] where x is Element of F₁() : P₁[x] } by RELAT_1:def 7; :: thesis: verum