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Theorem ballotlemelo 23046
Description: Elementhood in  O. (Contributed by Thierry Arnoux, 17-Apr-2017.)
Hypotheses
Ref Expression
ballotth.m  |-  M  e.  NN
ballotth.n  |-  N  e.  NN
ballotth.o  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
Assertion
Ref Expression
ballotlemelo  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
Distinct variable groups:    M, c    N, c    O, c
Allowed substitution hint:    C( c)

Proof of Theorem ballotlemelo
Dummy variable  d is distinct from all other variables.
StepHypRef Expression
1 fveq2 5525 . . . 4  |-  ( d  =  C  ->  ( # `
 d )  =  ( # `  C
) )
21eqeq1d 2291 . . 3  |-  ( d  =  C  ->  (
( # `  d )  =  M  <->  ( # `  C
)  =  M ) )
3 ballotth.o . . . 4  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 nfcv 2419 . . . . 5  |-  F/_ c ~P ( 1 ... ( M  +  N )
)
5 nfcv 2419 . . . . 5  |-  F/_ d ~P ( 1 ... ( M  +  N )
)
6 nfv 1605 . . . . 5  |-  F/ d ( # `  c
)  =  M
7 nfv 1605 . . . . 5  |-  F/ c ( # `  d
)  =  M
8 fveq2 5525 . . . . . 6  |-  ( c  =  d  ->  ( # `
 c )  =  ( # `  d
) )
98eqeq1d 2291 . . . . 5  |-  ( c  =  d  ->  (
( # `  c )  =  M  <->  ( # `  d
)  =  M ) )
104, 5, 6, 7, 9cbvrab 2786 . . . 4  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  =  { d  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  d )  =  M }
113, 10eqtri 2303 . . 3  |-  O  =  { d  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  d )  =  M }
122, 11elrab2 2925 . 2  |-  ( C  e.  O  <->  ( C  e.  ~P ( 1 ... ( M  +  N
) )  /\  ( # `
 C )  =  M ) )
13 elex 2796 . . . 4  |-  ( C  e.  ~P ( 1 ... ( M  +  N ) )  ->  C  e.  _V )
14 ovex 5883 . . . . 5  |-  ( 1 ... ( M  +  N ) )  e. 
_V
1514ssex 4158 . . . 4  |-  ( C 
C_  ( 1 ... ( M  +  N
) )  ->  C  e.  _V )
16 elpwg 3632 . . . 4  |-  ( C  e.  _V  ->  ( C  e.  ~P (
1 ... ( M  +  N ) )  <->  C  C_  (
1 ... ( M  +  N ) ) ) )
1713, 15, 16pm5.21nii 342 . . 3  |-  ( C  e.  ~P ( 1 ... ( M  +  N ) )  <->  C  C_  (
1 ... ( M  +  N ) ) )
1817anbi1i 676 . 2  |-  ( ( C  e.  ~P (
1 ... ( M  +  N ) )  /\  ( # `  C )  =  M )  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
1912, 18bitri 240 1  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    /\ wa 358    = wceq 1623    e. wcel 1684   {crab 2547   _Vcvv 2788    C_ wss 3152   ~Pcpw 3625   ` cfv 5255  (class class class)co 5858   1c1 8738    + caddc 8740   NNcn 9746   ...cfz 10782   #chash 11337
This theorem is referenced by:  ballotlemscr  23077  ballotlemro  23081  ballotlemfg  23084  ballotlemfrc  23085  ballotlemfrceq  23087  ballotlemrinv0  23091
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-iota 5219  df-fv 5263  df-ov 5861
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