Theorem elfix 32010

Description: Membership in the fixpoints of a class. (Contributed by Scott Fenton, 11-Apr-2012.)

Hypothesis

Ref	Expression
elfix.1	|- A e. _V

Assertion

Ref	Expression
elfix	|- ( A e. Fix R <-> A R A )

Proof of Theorem elfix

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fix 31966	. . 3 |- Fix R = dom ( R i^i _I )
2	1	eleq2i 2693	. 2 |- ( A e. Fix R <-> A e. dom ( R i^i _I ) )
3		elfix.1	. . . 4 |- A e. _V
4	3	eldm 5321	. . 3 |- ( A e. dom ( R i^i _I ) <-> E. x A ( R i^i _I ) x )
5		brin 4704	. . . . 5 |- ( A ( R i^i _I ) x <-> ( A R x /\ A _I x ) )
6		ancom 466	. . . . 5 |- ( ( A R x /\ A _I x ) <-> ( A _I x /\ A R x ) )
7		vex 3203	. . . . . . . 8 |- x e. _V
8	7	ideq 5274	. . . . . . 7 |- ( A _I x <-> A = x )
9		eqcom 2629	. . . . . . 7 |- ( A = x <-> x = A )
10	8, 9	bitri 264	. . . . . 6 |- ( A _I x <-> x = A )
11	10	anbi1i 731	. . . . 5 |- ( ( A _I x /\ A R x ) <-> ( x = A /\ A R x ) )
12	5, 6, 11	3bitri 286	. . . 4 |- ( A ( R i^i _I ) x <-> ( x = A /\ A R x ) )
13	12	exbii 1774	. . 3 |- ( E. x A ( R i^i _I ) x <-> E. x ( x = A /\ A R x ) )
14	4, 13	bitri 264	. 2 |- ( A e. dom ( R i^i _I ) <-> E. x ( x = A /\ A R x ) )
15		breq2 4657	. . 3 |- ( x = A -> ( A R x <-> A R A ) )
16	3, 15	ceqsexv 3242	. 2 |- ( E. x ( x = A /\ A R x ) <-> A R A )
17	2, 14, 16	3bitri 286	1 |- ( A e. Fix R <-> A R A )

Syntax hints:   <-> wb 196   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   _Vcvv 3200   i^i cin 3573   class class class wbr 4653   _I cid 5023   dom cdm 5114   Fixcfix 31942

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  ax-pr 4906

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-mo 2475  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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-dm 5124  df-fix 31966

This theorem is referenced by:  elfix2  32011  dffix2  32012  fixcnv  32015  ellimits  32017  elfuns  32022  dfrecs2  32057  dfrdg4  32058