Theorem ballotlemelo 27007

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 5792	. . . 4 |- ( d = C -> ( # ` d ) = ( # ` C ) )
2	1	eqeq1d 2453	. . 3 |- ( d = C -> ( ( # ` d ) = M <-> ( # ` C ) = M ) )
3		ballotth.o	. . . 4 |- O = { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M }
4		fveq2 5792	. . . . . 6 |- ( c = d -> ( # ` c ) = ( # ` d ) )
5	4	eqeq1d 2453	. . . . 5 |- ( c = d -> ( ( # ` c ) = M <-> ( # ` d ) = M ) )
6	5	cbvrabv 3070	. . . 4 |- { c e. ~P ( 1 ... ( M + N ) ) | ( # ` c ) = M } = { d e. ~P ( 1 ... ( M + N ) ) | ( # ` d ) = M }
7	3, 6	eqtri 2480	. . 3 |- O = { d e. ~P ( 1 ... ( M + N ) ) | ( # ` d ) = M }
8	2, 7	elrab2 3219	. 2 |- ( C e. O <-> ( C e. ~P ( 1 ... ( M + N ) ) /\ ( # ` C ) = M ) )
9		ovex 6218	. . . 4 |- ( 1 ... ( M + N ) ) e. _V
10	9	elpw2 4557	. . 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 1370   e. wcel 1758   {crab 2799   C_ wss 3429   ~Pcpw 3961   ` cfv 5519  (class class class)co 6193   1c1 9387   + caddc 9389   NNcn 10426   ...cfz 11547   #chash 12213

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-pw 3963  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-ov 6196

This theorem is referenced by:  ballotlemscr  27038  ballotlemro  27042  ballotlemfg  27045  ballotlemfrc  27046  ballotlemfrceq  27048  ballotlemrinv0  27052