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Theorem plybss 22885
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 22879 . . . 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 5321 . . 3  |-  dom Poly  C_  ~P CC
3 elfvdm 5877 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  e.  dom Poly )
42, 3sseldi 3442 . 2  |-  ( F  e.  (Poly `  S
)  ->  S  e.  ~P CC )
54elpwid 3967 1  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1407    e. wcel 1844   {cab 2389   E.wrex 2757    u. cun 3414    C_ wss 3416   ~Pcpw 3957   {csn 3974    |-> cmpt 4455   dom cdm 4825   ` cfv 5571  (class class class)co 6280    ^m cmap 7459   CCcc 9522   0cc0 9524    x. cmul 9529   NN0cn0 10838   ...cfz 11728   ^cexp 12212   sum_csu 13659  Polycply 22875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-8 1846  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pow 4574  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-pw 3959  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-mpt 4457  df-xp 4831  df-rel 4832  df-cnv 4833  df-dm 4835  df-rn 4836  df-res 4837  df-ima 4838  df-iota 5535  df-fv 5579  df-ply 22879
This theorem is referenced by:  elply  22886  plyf  22889  plyssc  22891  plyaddlem  22906  plymullem  22907  plysub  22910  dgrlem  22920  coeidlem  22928  plyco  22932  plycj  22968  plyreres  22973  plydivlem3  22985  plydivlem4  22986  elmnc  35462
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