Theorem ballotlemelo 28690

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 5848	. . . 4 |- ( d = C -> ( # ` d ) = ( # ` C ) )
2	1	eqeq1d 2456	. . 3 |- ( d = C -> ( ( # ` d ) = M <-> ( # ` C ) = M ) )
3		ballotth.o	. . . 4 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
4		fveq2 5848	. . . . . 6 |- ( c = d -> ( # ` c ) = ( # ` d ) )
5	4	eqeq1d 2456	. . . . 5 |- ( c = d -> ( ( # ` c ) = M <-> ( # ` d ) = M ) )
6	5	cbvrabv 3105	. . . 4 |- { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M } = { d e. ~P ( 1 ... ( M + N ) ) | ( # ` d ) = M }
7	3, 6	eqtri 2483	. . 3 |- O = { d e. ~P ( 1 ... ( M + N ) ) | ( # ` d ) = M }
8	2, 7	elrab2 3256	. 2 |- ( C e. O <-> ( C e. ~P ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )
9		ovex 6298	. . . 4 |- ( 1 ... ( M + N ) ) e. _V
10	9	elpw2 4601	. . 3 |- ( C e. ~P ( 1 ... ( M + N ) ) <-> C C_ ( 1 ... ( M + N ) ) )
11	10	anbi1i 693	. 2 |- ( ( C e. ~P ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) <-> ( C C_ ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )
12	8, 11	bitri 249	1 |- ( C e. O <-> ( C C_ ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )

Syntax hints:   <-> wb 184   /\ wa 367   = wceq 1398   e. wcel 1823   {crab 2808   C_ wss 3461   ~Pcpw 3999   ` cfv 5570  (class class class)co 6270   1c1 9482   + caddc 9484   NNcn 10531   ...cfz 11675   #chash 12387

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-iota 5534  df-fv 5578  df-ov 6273

This theorem is referenced by:  ballotlemscr  28721  ballotlemro  28725  ballotlemfg  28728  ballotlemfrc  28729  ballotlemfrceq  28731  ballotlemrinv0  28735