Theorem elfix 31180

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

Hypothesis

Ref	Expression
elfix.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
elfix	⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)

Proof of Theorem elfix

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fix 31135	. . 3 ⊢ Fix 𝑅 = dom (𝑅 ∩ I )
2	1	eleq2i 2680	. 2 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴 ∈ dom (𝑅 ∩ I ))
3		elfix.1	. . . 4 ⊢ 𝐴 ∈ V
4	3	eldm 5243	. . 3 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥)
5		brin 4634	. . . . 5 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥 ∧ 𝐴 I 𝑥))
6		ancom 465	. . . . 5 ⊢ ((𝐴𝑅𝑥 ∧ 𝐴 I 𝑥) ↔ (𝐴 I 𝑥 ∧ 𝐴𝑅𝑥))
7		vex 3176	. . . . . . . 8 ⊢ 𝑥 ∈ V
8	7	ideq 5196	. . . . . . 7 ⊢ (𝐴 I 𝑥 ↔ 𝐴 = 𝑥)
9		eqcom 2617	. . . . . . 7 ⊢ (𝐴 = 𝑥 ↔ 𝑥 = 𝐴)
10	8, 9	bitri 263	. . . . . 6 ⊢ (𝐴 I 𝑥 ↔ 𝑥 = 𝐴)
11	10	anbi1i 727	. . . . 5 ⊢ ((𝐴 I 𝑥 ∧ 𝐴𝑅𝑥) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
12	5, 6, 11	3bitri 285	. . . 4 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
13	12	exbii 1764	. . 3 ⊢ (∃𝑥 𝐴(𝑅 ∩ I )𝑥 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
14	4, 13	bitri 263	. 2 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
15		breq2 4587	. . 3 ⊢ (𝑥 = 𝐴 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐴))
16	3, 15	ceqsexv 3215	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥) ↔ 𝐴𝑅𝐴)
17	2, 14, 16	3bitri 285	1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)

Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   class class class wbr 4583   I cid 4948  dom cdm 5038   Fix cfix 31111

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-dm 5048  df-fix 31135

This theorem is referenced by:  elfix2  31181  dffix2  31182  fixcnv  31185  ellimits  31187  elfuns  31192  dfrecs2  31227  dfrdg4  31228