Theorem ballotfilemelo 13200

Description: Elementhood in O. (Contributed by Thierry Arnoux, 17-Apr-2017.)

Hypotheses

Ref	Expression
ballotth.m	|- M e. NN
ballotth.n	|- N e. NN
ballotfilem.o	|- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }

Assertion

Ref	Expression
ballotfilemelo	|- ( C e. O <-> ( C C_ ( 1 ... ( M + N ) ) /\ C e. Fin /\ (♯ ` C ) = M ) )

Distinct variable groups:   M, c   N, c   O, c

Allowed substitution hint:   C( c)

Proof of Theorem ballotfilemelo

Dummy variable d is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfpw 7252	. . 3 |- ( C e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) <-> ( C C_ ( 1 ... ( M + N ) ) /\ C e. Fin ) )
2	1	anbi1i 462	. 2 |- ( ( C e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) /\ (♯ ` C ) = M ) <-> ( ( C C_ ( 1 ... ( M + N ) ) /\ C e. Fin ) /\ (♯ ` C ) = M ) )
3		fveqeq2 5699	. . 3 |- ( d = C -> ( (♯ ` d ) = M <-> (♯ ` C ) = M ) )
4		ballotfilem.o	. . . 4 |- O = { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M }
5		fveqeq2 5699	. . . . 5 |- ( c = d -> ( (♯ ` c ) = M <-> (♯ ` d ) = M ) )
6	5	cbvrabv 2820	. . . 4 |- { c e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` c ) = M } = { d e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` d ) = M }
7	4, 6	eqtri 2259	. . 3 |- O = { d e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` d ) = M }
8	3, 7	elrab2 2985	. 2 |- ( C e. O <-> ( C e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) /\ (♯ ` C ) = M ) )
9		df-3an 1011	. 2 |- ( ( C C_ ( 1 ... ( M + N ) ) /\ C e. Fin /\ (♯ ` C ) = M ) <-> ( ( C C_ ( 1 ... ( M + N ) ) /\ C e. Fin ) /\ (♯ ` C ) = M ) )
10	2, 8, 9	3bitr4i 212	1 |- ( C e. O <-> ( C C_ ( 1 ... ( M + N ) ) /\ C e. Fin /\ (♯ ` C ) = M ) )

Syntax hints:   /\ wa 104   <-> wb 105   /\ w3a 1009   = wceq 1402   e. wcel 2209   {crab 2532   i^i cin 3219   C_ wss 3220   ~Pcpw 3685   ` cfv 5372  (class class class)co 6075   Fincfn 7012   1c1 8170   + caddc 8172   NNcn 9283   ...cfz 10390  ♯chash 11192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-rex 2534  df-rab 2537  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-br 4126  df-iota 5332  df-fv 5380

This theorem is referenced by:  ballotfilemcdc  13201  ballotfilemfc0  13210  ballotfilemscr  13240  ballotfilemro  13244  ballotfilemrinv0  13254