set h = <-> f;

A1: dom (<-> f) = dom f by Def33;

rng (<-> f) c= Q_PFuncs (DOMS Y)

A1: dom (<-> f) = dom f by Def33;

rng (<-> f) c= Q_PFuncs (DOMS Y)

proof

hence
<-> f is PartFunc of X,(Q_PFuncs (DOMS Y))
by A1, RELSET_1:4; :: thesis: verum
let y be object ; :: according to TARSKI:def 3 :: thesis: ( not y in rng (<-> f) or y in Q_PFuncs (DOMS Y) )

assume y in rng (<-> f) ; :: thesis: y in Q_PFuncs (DOMS Y)

then consider x being object such that

A2: x in dom (<-> f) and

A3: (<-> f) . x = y by FUNCT_1:def 3;

reconsider y = y as Function by A3;

A4: (<-> f) . x = - (f . x) by A2, Def33;

A5: rng y c= RAT by A3, A4, RAT_1:def 2;

f . x in Y by A1, A2, PARTFUN1:4;

then dom (f . x) in { (dom m) where m is Element of Y : verum } ;

then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;

dom y = dom (f . x) by A3, A4, VALUED_1:8;

then y is PartFunc of (DOMS Y),RAT by A6, A5, RELSET_1:4;

hence y in Q_PFuncs (DOMS Y) by Def14; :: thesis: verum

end;assume y in rng (<-> f) ; :: thesis: y in Q_PFuncs (DOMS Y)

then consider x being object such that

A2: x in dom (<-> f) and

A3: (<-> f) . x = y by FUNCT_1:def 3;

reconsider y = y as Function by A3;

A4: (<-> f) . x = - (f . x) by A2, Def33;

A5: rng y c= RAT by A3, A4, RAT_1:def 2;

f . x in Y by A1, A2, PARTFUN1:4;

then dom (f . x) in { (dom m) where m is Element of Y : verum } ;

then A6: dom (f . x) c= DOMS Y by ZFMISC_1:74;

dom y = dom (f . x) by A3, A4, VALUED_1:8;

then y is PartFunc of (DOMS Y),RAT by A6, A5, RELSET_1:4;

hence y in Q_PFuncs (DOMS Y) by Def14; :: thesis: verum