Theorem ballotfilemelo 13166

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 7228	. . 3 |- ( C e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) <-> ( C C_ ( 1 ... ( M + N ) ) /\ C e. Fin ) )
2	1	anbi1i 458	. 2 |- ( ( C e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) /\ (♯ ` C ) = M ) <-> ( ( C C_ ( 1 ... ( M + N ) ) /\ C e. Fin ) /\ (♯ ` C ) = M ) )
3		fveqeq2 5684	. . 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 5684	. . . . 5 |- ( c = d -> ( (♯ ` c ) = M <-> (♯ ` d ) = M ) )
6	5	cbvrabv 2814	. . . 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 2255	. . 3 |- O = { d e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) | (♯ ` d ) = M }
8	3, 7	elrab2 2979	. 2 |- ( C e. O <-> ( C e. ( ~P ( 1 ... ( M + N ) ) i^i Fin ) /\ (♯ ` C ) = M ) )
9		df-3an 1007	. 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 1005   = wceq 1398   e. wcel 2205   {crab 2526   i^i cin 3213   C_ wss 3214   ~Pcpw 3674   ` cfv 5357  (class class class)co 6058   Fincfn 6988   1c1 8144   + caddc 8146   NNcn 9254   ...cfz 10361  ♯chash 11163

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-rex 2528  df-rab 2531  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-iota 5317  df-fv 5365

This theorem is referenced by:  ballotfilemcdc  13167  ballotfilemfc0  13176  ballotfilemscr  13206  ballotfilemro  13210  ballotfilemrinv0  13220