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Theorem plybss 22326
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 22320 . . . 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 5501 . . 3  |-  dom Poly  C_  ~P CC
3 elfvdm 5890 . . 3  |-  ( F  e.  (Poly `  S
)  ->  S  e.  dom Poly )
42, 3sseldi 3502 . 2  |-  ( F  e.  (Poly `  S
)  ->  S  e.  ~P CC )
54elpwid 4020 1  |-  ( F  e.  (Poly `  S
)  ->  S  C_  CC )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {cab 2452   E.wrex 2815    u. cun 3474    C_ wss 3476   ~Pcpw 4010   {csn 4027    |-> cmpt 4505   dom cdm 4999   ` cfv 5586  (class class class)co 6282    ^m cmap 7417   CCcc 9486   0cc0 9488    x. cmul 9493   NN0cn0 10791   ...cfz 11668   ^cexp 12130   sum_csu 13467  Polycply 22316
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-xp 5005  df-rel 5006  df-cnv 5007  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fv 5594  df-ply 22320
This theorem is referenced by:  elply  22327  plyf  22330  plyssc  22332  plyaddlem  22347  plymullem  22348  plysub  22351  dgrlem  22361  coeidlem  22369  plyco  22373  plycj  22408  plyreres  22413  plydivlem3  22425  plydivlem4  22426  elmnc  30690
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