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Theorem ballotlemelo 23919
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 5563 . . . 4  |-  ( d  =  C  ->  ( # `
 d )  =  ( # `  C
) )
21eqeq1d 2324 . . 3  |-  ( d  =  C  ->  (
( # `  d )  =  M  <->  ( # `  C
)  =  M ) )
3 ballotth.o . . . 4  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 nfcv 2452 . . . . 5  |-  F/_ c ~P ( 1 ... ( M  +  N )
)
5 nfcv 2452 . . . . 5  |-  F/_ d ~P ( 1 ... ( M  +  N )
)
6 nfv 1610 . . . . 5  |-  F/ d ( # `  c
)  =  M
7 nfv 1610 . . . . 5  |-  F/ c ( # `  d
)  =  M
8 fveq2 5563 . . . . . 6  |-  ( c  =  d  ->  ( # `
 c )  =  ( # `  d
) )
98eqeq1d 2324 . . . . 5  |-  ( c  =  d  ->  (
( # `  c )  =  M  <->  ( # `  d
)  =  M ) )
104, 5, 6, 7, 9cbvrab 2820 . . . 4  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  =  { d  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  d )  =  M }
113, 10eqtri 2336 . . 3  |-  O  =  { d  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  d )  =  M }
122, 11elrab2 2959 . 2  |-  ( C  e.  O  <->  ( C  e.  ~P ( 1 ... ( M  +  N
) )  /\  ( # `
 C )  =  M ) )
13 elex 2830 . . . 4  |-  ( C  e.  ~P ( 1 ... ( M  +  N ) )  ->  C  e.  _V )
14 ovex 5925 . . . . 5  |-  ( 1 ... ( M  +  N ) )  e. 
_V
1514ssex 4195 . . . 4  |-  ( C 
C_  ( 1 ... ( M  +  N
) )  ->  C  e.  _V )
16 elpwg 3666 . . . 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 1633    e. wcel 1701   {crab 2581   _Vcvv 2822    C_ wss 3186   ~Pcpw 3659   ` cfv 5292  (class class class)co 5900   1c1 8783    + caddc 8785   NNcn 9791   ...cfz 10829   #chash 11384
This theorem is referenced by:  ballotlemscr  23950  ballotlemro  23954  ballotlemfg  23957  ballotlemfrc  23958  ballotlemfrceq  23960  ballotlemrinv0  23964
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-pw 3661  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-iota 5256  df-fv 5300  df-ov 5903
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