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Theorem ballotlemelo 27007
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 5792 . . . 4  |-  ( d  =  C  ->  ( # `
 d )  =  ( # `  C
) )
21eqeq1d 2453 . . 3  |-  ( d  =  C  ->  (
( # `  d )  =  M  <->  ( # `  C
)  =  M ) )
3 ballotth.o . . . 4  |-  O  =  { c  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  c )  =  M }
4 fveq2 5792 . . . . . 6  |-  ( c  =  d  ->  ( # `
 c )  =  ( # `  d
) )
54eqeq1d 2453 . . . . 5  |-  ( c  =  d  ->  (
( # `  c )  =  M  <->  ( # `  d
)  =  M ) )
65cbvrabv 3070 . . . 4  |-  { c  e.  ~P ( 1 ... ( M  +  N ) )  |  ( # `  c
)  =  M }  =  { d  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  d )  =  M }
73, 6eqtri 2480 . . 3  |-  O  =  { d  e.  ~P ( 1 ... ( M  +  N )
)  |  ( # `  d )  =  M }
82, 7elrab2 3219 . 2  |-  ( C  e.  O  <->  ( C  e.  ~P ( 1 ... ( M  +  N
) )  /\  ( # `
 C )  =  M ) )
9 ovex 6218 . . . 4  |-  ( 1 ... ( M  +  N ) )  e. 
_V
109elpw2 4557 . . 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 1370    e. wcel 1758   {crab 2799    C_ wss 3429   ~Pcpw 3961   ` cfv 5519  (class class class)co 6193   1c1 9387    + caddc 9389   NNcn 10426   ...cfz 11547   #chash 12213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-ov 6196
This theorem is referenced by:  ballotlemscr  27038  ballotlemro  27042  ballotlemfg  27045  ballotlemfrc  27046  ballotlemfrceq  27048  ballotlemrinv0  27052
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