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Theorem ballotlemelo 24745
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 5728 . . . 4  |-  ( d  =  C  ->  ( # `
 d )  =  ( # `  C
) )
21eqeq1d 2444 . . 3  |-  ( d  =  C  ->  (
( # `  d )  =  M  <->  ( # `  C
)  =  M ) )
3 ballotth.o . . . 4  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 fveq2 5728 . . . . . 6  |-  ( c  =  d  ->  ( # `
 c )  =  ( # `  d
) )
54eqeq1d 2444 . . . . 5  |-  ( c  =  d  ->  (
( # `  c )  =  M  <->  ( # `  d
)  =  M ) )
65cbvrabv 2955 . . . 4  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  =  { d  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  d )  =  M }
73, 6eqtri 2456 . . 3  |-  O  =  { d  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  d )  =  M }
82, 7elrab2 3094 . 2  |-  ( C  e.  O  <->  ( C  e.  ~P ( 1 ... ( M  +  N
) )  /\  ( # `
 C )  =  M ) )
9 ovex 6106 . . . 4  |-  ( 1 ... ( M  +  N ) )  e. 
_V
109elpw2 4364 . . 3  |-  ( C  e.  ~P ( 1 ... ( M  +  N ) )  <->  C  C_  (
1 ... ( M  +  N ) ) )
1110anbi1i 677 . 2  |-  ( ( C  e.  ~P (
1 ... ( M  +  N ) )  /\  ( # `  C )  =  M )  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
128, 11bitri 241 1  |-  ( C  e.  O  <->  ( C  C_  ( 1 ... ( M  +  N )
)  /\  ( # `  C
)  =  M ) )
Colors of variables: wff set class
Syntax hints:    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725   {crab 2709    C_ wss 3320   ~Pcpw 3799   ` cfv 5454  (class class class)co 6081   1c1 8991    + caddc 8993   NNcn 10000   ...cfz 11043   #chash 11618
This theorem is referenced by:  ballotlemscr  24776  ballotlemro  24780  ballotlemfg  24783  ballotlemfrc  24784  ballotlemfrceq  24786  ballotlemrinv0  24790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-iota 5418  df-fv 5462  df-ov 6084
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