Theorem ballotlemelo 26822

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 5686	. . . 4 |- ( d = C -> ( # ` d ) = ( # ` C ) )
2	1	eqeq1d 2446	. . 3 |- ( d = C -> ( ( # ` d ) = M <-> ( # ` C ) = M ) )
3		ballotth.o	. . . 4 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
4		fveq2 5686	. . . . . 6 |- ( c = d -> ( # ` c ) = ( # ` d ) )
5	4	eqeq1d 2446	. . . . 5 |- ( c = d -> ( ( # ` c ) = M <-> ( # ` d ) = M ) )
6	5	cbvrabv 2966	. . . 4 |- { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M } = { d e. ~P ( 1 ... ( M + N ) ) | ( # ` d ) = M }
7	3, 6	eqtri 2458	. . 3 |- O = { d e. ~P ( 1 ... ( M + N ) ) | ( # ` d ) = M }
8	2, 7	elrab2 3114	. 2 |- ( C e. O <-> ( C e. ~P ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )
9		ovex 6111	. . . 4 |- ( 1 ... ( M + N ) ) e. _V
10	9	elpw2 4451	. . 3 |- ( C e. ~P ( 1 ... ( M + N ) ) <-> C C_ ( 1 ... ( M + N ) ) )
11	10	anbi1i 695	. 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 369   = wceq 1369   e. wcel 1756   {crab 2714   C_ wss 3323   ~Pcpw 3855   ` cfv 5413  (class class class)co 6086   1c1 9275   + caddc 9277   NNcn 10314   ...cfz 11429   #chash 12095

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-iota 5376  df-fv 5421  df-ov 6089

This theorem is referenced by:  ballotlemscr  26853  ballotlemro  26857  ballotlemfg  26860  ballotlemfrc  26861  ballotlemfrceq  26863  ballotlemrinv0  26867