Theorem elfix 36401

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 36357	. . 3 ⊢ Fix 𝑅 = dom (𝑅 ∩ I )
2	1	eleq2i 2854	. 2 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴 ∈ dom (𝑅 ∩ I ))
3		elfix.1	. . . 4 ⊢ 𝐴 ∈ V
4	3	eldm 5889	. . 3 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥)
5		brin 5162	. . . . 5 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥 ∧ 𝐴 I 𝑥))
6		ancom 465	. . . . 5 ⊢ ((𝐴𝑅𝑥 ∧ 𝐴 I 𝑥) ↔ (𝐴 I 𝑥 ∧ 𝐴𝑅𝑥))
7		vex 3458	. . . . . . . 8 ⊢ 𝑥 ∈ V
8	7	ideq 5837	. . . . . . 7 ⊢ (𝐴 I 𝑥 ↔ 𝐴 = 𝑥)
9		eqcom 2769	. . . . . . 7 ⊢ (𝐴 = 𝑥 ↔ 𝑥 = 𝐴)
10	8, 9	bitri 278	. . . . . 6 ⊢ (𝐴 I 𝑥 ↔ 𝑥 = 𝐴)
11	10	anbi1i 635	. . . . 5 ⊢ ((𝐴 I 𝑥 ∧ 𝐴𝑅𝑥) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
12	5, 6, 11	3bitri 300	. . . 4 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
13	12	exbii 1877	. . 3 ⊢ (∃𝑥 𝐴(𝑅 ∩ I )𝑥 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
14	4, 13	bitri 278	. 2 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
15		breq2 5112	. . 3 ⊢ (𝑥 = 𝐴 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐴))
16	3, 15	ceqsexv 3502	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥) ↔ 𝐴𝑅𝐴)
17	2, 14, 16	3bitri 300	1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 400   = wceq 1569  ∃wex 1808   ∈ wcel 2142  Vcvv 3454   ∩ cin 3903   class class class wbr 5108   I cid 5554  dom cdm 5660   Fix cfix 36333

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-pr 5403

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-id 5555  df-xp 5666  df-rel 5667  df-dm 5670  df-fix 36357

This theorem is used by:  elfix2  36402  dffix2  36403  fixcnv  36406  ellimits  36408  elfuns  36413  dfrecs2  36450  dfrdg4  36451