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Theorem ballotlemelo 28690
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 5848 . . . 4  |-  ( d  =  C  ->  ( # `
 d )  =  ( # `  C
) )
21eqeq1d 2456 . . 3  |-  ( d  =  C  ->  (
( # `  d )  =  M  <->  ( # `  C
)  =  M ) )
3 ballotth.o . . . 4  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 fveq2 5848 . . . . . 6  |-  ( c  =  d  ->  ( # `
 c )  =  ( # `  d
) )
54eqeq1d 2456 . . . . 5  |-  ( c  =  d  ->  (
( # `  c )  =  M  <->  ( # `  d
)  =  M ) )
65cbvrabv 3105 . . . 4  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  =  { d  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  d )  =  M }
73, 6eqtri 2483 . . 3  |-  O  =  { d  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  d )  =  M }
82, 7elrab2 3256 . 2  |-  ( C  e.  O  <->  ( C  e.  ~P ( 1 ... ( M  +  N
) )  /\  ( # `
 C )  =  M ) )
9 ovex 6298 . . . 4  |-  ( 1 ... ( M  +  N ) )  e. 
_V
109elpw2 4601 . . 3  |-  ( C  e.  ~P ( 1 ... ( M  +  N ) )  <->  C  C_  (
1 ... ( M  +  N ) ) )
1110anbi1i 693 . 2  |-  ( ( C  e.  ~P (
1 ... ( M  +  N ) )  /\  ( # `  C )  =  M )  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
128, 11bitri 249 1  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   {crab 2808    C_ wss 3461   ~Pcpw 3999   ` cfv 5570  (class class class)co 6270   1c1 9482    + caddc 9484   NNcn 10531   ...cfz 11675   #chash 12387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273
This theorem is referenced by:  ballotlemscr  28721  ballotlemro  28725  ballotlemfg  28728  ballotlemfrc  28729  ballotlemfrceq  28731  ballotlemrinv0  28735
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