Theorem ballotlemelo 30549

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.

Step	Hyp	Ref	Expression
1		fveq2 6191	. . . 4 |- ( d = C -> ( # ` d ) = ( # ` C ) )
2	1	eqeq1d 2624	. . 3 |- ( d = C -> ( ( # ` d ) = M <-> ( # ` C ) = M ) )
3		ballotth.o	. . . 4 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
4		fveq2 6191	. . . . . 6 |- ( c = d -> ( # ` c ) = ( # ` d ) )
5	4	eqeq1d 2624	. . . . 5 |- ( c = d -> ( ( # ` c ) = M <-> ( # ` d ) = M ) )
6	5	cbvrabv 3199	. . . 4 |- { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M } = { d e. ~P ( 1 ... ( M + N ) ) | ( # ` d ) = M }
7	3, 6	eqtri 2644	. . 3 |- O = { d e. ~P ( 1 ... ( M + N ) ) | ( # ` d ) = M }
8	2, 7	elrab2 3366	. 2 |- ( C e. O <-> ( C e. ~P ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )
9		ovex 6678	. . . 4 |- ( 1 ... ( M + N ) ) e. _V
10	9	elpw2 4828	. . 3 |- ( C e. ~P ( 1 ... ( M + N ) ) <-> C C_ ( 1 ... ( M + N ) ) )
11	10	anbi1i 731	. 2 |- ( ( C e. ~P ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) <-> ( C C_ ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )
12	8, 11	bitri 264	1 |- ( C e. O <-> ( C C_ ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   {crab 2916   C_ wss 3574   ~Pcpw 4158   ` cfv 5888  (class class class)co 6650   1c1 9937   + caddc 9939   NNcn 11020   ...cfz 12326   #chash 13117

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-ov 6653

This theorem is referenced by:  ballotlemscr  30580  ballotlemro  30584  ballotlemfg  30587  ballotlemfrc  30588  ballotlemfrceq  30590  ballotlemrinv0  30594