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Theorem ballotlemelo 26822
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 5686 . . . 4  |-  ( d  =  C  ->  ( # `
 d )  =  ( # `  C
) )
21eqeq1d 2446 . . 3  |-  ( d  =  C  ->  (
( # `  d )  =  M  <->  ( # `  C
)  =  M ) )
3 ballotth.o . . . 4  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 fveq2 5686 . . . . . 6  |-  ( c  =  d  ->  ( # `
 c )  =  ( # `  d
) )
54eqeq1d 2446 . . . . 5  |-  ( c  =  d  ->  (
( # `  c )  =  M  <->  ( # `  d
)  =  M ) )
65cbvrabv 2966 . . . 4  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  =  { d  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  d )  =  M }
73, 6eqtri 2458 . . 3  |-  O  =  { d  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  d )  =  M }
82, 7elrab2 3114 . 2  |-  ( C  e.  O  <->  ( C  e.  ~P ( 1 ... ( M  +  N
) )  /\  ( # `
 C )  =  M ) )
9 ovex 6111 . . . 4  |-  ( 1 ... ( M  +  N ) )  e. 
_V
109elpw2 4451 . . 3  |-  ( C  e.  ~P ( 1 ... ( M  +  N ) )  <->  C  C_  (
1 ... ( M  +  N ) ) )
1110anbi1i 695 . 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 369    = wceq 1369    e. wcel 1756   {crab 2714    C_ wss 3323   ~Pcpw 3855   ` cfv 5413  (class class class)co 6086   1c1 9275    + caddc 9277   NNcn 10314   ...cfz 11429   #chash 12095
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-iota 5376  df-fv 5421  df-ov 6089
This theorem is referenced by:  ballotlemscr  26853  ballotlemro  26857  ballotlemfg  26860  ballotlemfrc  26861  ballotlemfrceq  26863  ballotlemrinv0  26867
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