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Theorem plybss 21780
Description: Reverse closure of the parameter  S of the polynomial set function. (Contributed by Mario Carneiro, 22-Jul-2014.)
Assertion
Ref Expression
plybss  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )

Proof of Theorem plybss
Dummy variables  k 
a  n  z  f  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ply 21774 . . . 4  |- Poly  =  ( x  e.  ~P CC  |->  { f  |  E. n  e.  NN0  E. a  e.  ( ( x  u. 
{ 0 } )  ^m  NN0 ) f  =  ( z  e.  CC  |->  sum_ k  e.  ( 0 ... n ) ( ( a `  k )  x.  (
z ^ k ) ) ) } )
21dmmptss 5434 . . 3  |-  dom Poly  C_  ~P CC
3 elfvdm 5817 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  e.  dom Poly )
42, 3sseldi 3454 . 2  |-  ( F  e.  (Poly `  S
)  ->  S  e.  ~P CC )
54elpwid 3970 1  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    e. wcel 1758   {cab 2436   E.wrex 2796    u. cun 3426    C_ wss 3428   ~Pcpw 3960   {csn 3977    |-> cmpt 4450   dom cdm 4940   ` cfv 5518  (class class class)co 6192    ^m cmap 7316   CCcc 9383   0cc0 9385    x. cmul 9390   NN0cn0 10682   ...cfz 11540   ^cexp 11968   sum_csu 13267  Polycply 21770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4513  ax-nul 4521  ax-pow 4570  ax-pr 4631
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3072  df-dif 3431  df-un 3433  df-in 3435  df-ss 3442  df-nul 3738  df-if 3892  df-pw 3962  df-sn 3978  df-pr 3980  df-op 3984  df-uni 4192  df-br 4393  df-opab 4451  df-mpt 4452  df-xp 4946  df-rel 4947  df-cnv 4948  df-dm 4950  df-rn 4951  df-res 4952  df-ima 4953  df-iota 5481  df-fv 5526  df-ply 21774
This theorem is referenced by:  elply  21781  plyf  21784  plyssc  21786  plyaddlem  21801  plymullem  21802  plysub  21805  dgrlem  21815  coeidlem  21823  plyco  21827  plycj  21862  plyreres  21867  plydivlem3  21879  plydivlem4  21880  elmnc  29633
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