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Theorem ply1frcl 18491
Description: Reverse closure for the set of univariate polynomial functions. (Contributed by AV, 9-Sep-2019.)
Hypothesis
Ref Expression
ply1frcl.q  |-  Q  =  ran  ( S evalSub1  R )
Assertion
Ref Expression
ply1frcl  |-  ( X  e.  Q  ->  ( S  e.  _V  /\  R  e.  ~P ( Base `  S
) ) )

Proof of Theorem ply1frcl
Dummy variables  r 
b  s  x  y  e are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ne0i 3734 . . 3  |-  ( X  e.  ran  ( S evalSub1  R )  ->  ran  ( S evalSub1  R )  =/=  (/) )
2 ply1frcl.q . . 3  |-  Q  =  ran  ( S evalSub1  R )
31, 2eleq2s 2504 . 2  |-  ( X  e.  Q  ->  ran  ( S evalSub1  R )  =/=  (/) )
4 rneq 5158 . . . 4  |-  ( ( S evalSub1  R )  =  (/)  ->  ran  ( S evalSub1  R )  =  ran  (/) )
5 rn0 5184 . . . 4  |-  ran  (/)  =  (/)
64, 5syl6eq 2453 . . 3  |-  ( ( S evalSub1  R )  =  (/)  ->  ran  ( S evalSub1  R )  =  (/) )
76necon3i 2636 . 2  |-  ( ran  ( S evalSub1  R )  =/=  (/)  ->  ( S evalSub1  R )  =/=  (/) )
8 n0 3738 . . 3  |-  ( ( S evalSub1  R )  =/=  (/)  <->  E. e 
e  e.  ( S evalSub1  R ) )
9 df-evls1 18488 . . . . . . 7  |- evalSub1  =  ( s  e.  _V ,  r  e. 
~P ( Base `  s
)  |->  [_ ( Base `  s
)  /  b ]_ ( ( x  e.  ( b  ^m  (
b  ^m  1o )
)  |->  ( x  o.  ( y  e.  b 
|->  ( 1o  X.  {
y } ) ) ) )  o.  (
( 1o evalSub  s ) `  r ) ) )
109dmmpt2ssx 6786 . . . . . 6  |-  dom evalSub1  C_  U_ s  e.  _V  ( { s }  X.  ~P ( Base `  s ) )
11 elfvdm 5817 . . . . . . 7  |-  ( e  e.  ( evalSub1  `  <. S ,  R >. )  ->  <. S ,  R >.  e.  dom evalSub1  )
12 df-ov 6221 . . . . . . 7  |-  ( S evalSub1  R )  =  ( evalSub1  `  <. S ,  R >. )
1311, 12eleq2s 2504 . . . . . 6  |-  ( e  e.  ( S evalSub1  R )  ->  <. S ,  R >.  e.  dom evalSub1  )
1410, 13sseldi 3432 . . . . 5  |-  ( e  e.  ( S evalSub1  R )  ->  <. S ,  R >.  e.  U_ s  e. 
_V  ( { s }  X.  ~P ( Base `  s ) ) )
15 fveq2 5791 . . . . . . 7  |-  ( s  =  S  ->  ( Base `  s )  =  ( Base `  S
) )
1615pweqd 3949 . . . . . 6  |-  ( s  =  S  ->  ~P ( Base `  s )  =  ~P ( Base `  S
) )
1716opeliunxp2 5071 . . . . 5  |-  ( <. S ,  R >.  e. 
U_ s  e.  _V  ( { s }  X.  ~P ( Base `  s
) )  <->  ( S  e.  _V  /\  R  e. 
~P ( Base `  S
) ) )
1814, 17sylib 196 . . . 4  |-  ( e  e.  ( S evalSub1  R )  ->  ( S  e. 
_V  /\  R  e.  ~P ( Base `  S
) ) )
1918exlimiv 1737 . . 3  |-  ( E. e  e  e.  ( S evalSub1  R )  ->  ( S  e.  _V  /\  R  e.  ~P ( Base `  S
) ) )
208, 19sylbi 195 . 2  |-  ( ( S evalSub1  R )  =/=  (/)  ->  ( S  e.  _V  /\  R  e.  ~P ( Base `  S
) ) )
213, 7, 203syl 20 1  |-  ( X  e.  Q  ->  ( S  e.  _V  /\  R  e.  ~P ( Base `  S
) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 367    = wceq 1399   E.wex 1627    e. wcel 1836    =/= wne 2591   _Vcvv 3051   [_csb 3365   (/)c0 3728   ~Pcpw 3944   {csn 3961   <.cop 3967   U_ciun 4260    |-> cmpt 4442    X. cxp 4928   dom cdm 4930   ran crn 4931    o. ccom 4934   ` cfv 5513  (class class class)co 6218   1oc1o 7063    ^m cmap 7360   Basecbs 14657   evalSub ces 18305   evalSub1 ces1 18486
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1633  ax-4 1646  ax-5 1719  ax-6 1765  ax-7 1808  ax-8 1838  ax-9 1840  ax-10 1855  ax-11 1860  ax-12 1872  ax-13 2020  ax-ext 2374  ax-sep 4505  ax-nul 4513  ax-pow 4560  ax-pr 4618  ax-un 6513
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1402  df-ex 1628  df-nf 1632  df-sb 1758  df-eu 2236  df-mo 2237  df-clab 2382  df-cleq 2388  df-clel 2391  df-nfc 2546  df-ne 2593  df-ral 2751  df-rex 2752  df-rab 2755  df-v 3053  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3729  df-if 3875  df-pw 3946  df-sn 3962  df-pr 3964  df-op 3968  df-uni 4181  df-iun 4262  df-br 4385  df-opab 4443  df-mpt 4444  df-id 4726  df-xp 4936  df-rel 4937  df-cnv 4938  df-co 4939  df-dm 4940  df-rn 4941  df-res 4942  df-ima 4943  df-iota 5477  df-fun 5515  df-fv 5521  df-ov 6221  df-oprab 6222  df-mpt2 6223  df-1st 6721  df-2nd 6722  df-evls1 18488
This theorem is referenced by: (None)
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