Theorem elfix 35966

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 35922	. . 3 ⊢ Fix 𝑅 = dom (𝑅 ∩ I )
2	1	eleq2i 2825	. 2 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴 ∈ dom (𝑅 ∩ I ))
3		elfix.1	. . . 4 ⊢ 𝐴 ∈ V
4	3	eldm 5844	. . 3 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥)
5		brin 5145	. . . . 5 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥 ∧ 𝐴 I 𝑥))
6		ancom 460	. . . . 5 ⊢ ((𝐴𝑅𝑥 ∧ 𝐴 I 𝑥) ↔ (𝐴 I 𝑥 ∧ 𝐴𝑅𝑥))
7		vex 3441	. . . . . . . 8 ⊢ 𝑥 ∈ V
8	7	ideq 5796	. . . . . . 7 ⊢ (𝐴 I 𝑥 ↔ 𝐴 = 𝑥)
9		eqcom 2740	. . . . . . 7 ⊢ (𝐴 = 𝑥 ↔ 𝑥 = 𝐴)
10	8, 9	bitri 275	. . . . . 6 ⊢ (𝐴 I 𝑥 ↔ 𝑥 = 𝐴)
11	10	anbi1i 624	. . . . 5 ⊢ ((𝐴 I 𝑥 ∧ 𝐴𝑅𝑥) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
12	5, 6, 11	3bitri 297	. . . 4 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
13	12	exbii 1849	. . 3 ⊢ (∃𝑥 𝐴(𝑅 ∩ I )𝑥 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
14	4, 13	bitri 275	. 2 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
15		breq2 5097	. . 3 ⊢ (𝑥 = 𝐴 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐴))
16	3, 15	ceqsexv 3487	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥) ↔ 𝐴𝑅𝐴)
17	2, 14, 16	3bitri 297	1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1541  ∃wex 1780   ∈ wcel 2113  Vcvv 3437   ∩ cin 3897   class class class wbr 5093   I cid 5513  dom cdm 5619   Fix cfix 35898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-sn 4576  df-pr 4578  df-op 4582  df-br 5094  df-opab 5156  df-id 5514  df-xp 5625  df-rel 5626  df-dm 5629  df-fix 35922

This theorem is referenced by:  elfix2  35967  dffix2  35968  fixcnv  35971  ellimits  35973  elfuns  35978  dfrecs2  36015  dfrdg4  36016