Theorem elfix 33357

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 33313	. . 3 ⊢ Fix 𝑅 = dom (𝑅 ∩ I )
2	1	eleq2i 2902	. 2 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴 ∈ dom (𝑅 ∩ I ))
3		elfix.1	. . . 4 ⊢ 𝐴 ∈ V
4	3	eldm 5762	. . 3 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥)
5		brin 5109	. . . . 5 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥 ∧ 𝐴 I 𝑥))
6		ancom 463	. . . . 5 ⊢ ((𝐴𝑅𝑥 ∧ 𝐴 I 𝑥) ↔ (𝐴 I 𝑥 ∧ 𝐴𝑅𝑥))
7		vex 3496	. . . . . . . 8 ⊢ 𝑥 ∈ V
8	7	ideq 5716	. . . . . . 7 ⊢ (𝐴 I 𝑥 ↔ 𝐴 = 𝑥)
9		eqcom 2826	. . . . . . 7 ⊢ (𝐴 = 𝑥 ↔ 𝑥 = 𝐴)
10	8, 9	bitri 277	. . . . . 6 ⊢ (𝐴 I 𝑥 ↔ 𝑥 = 𝐴)
11	10	anbi1i 625	. . . . 5 ⊢ ((𝐴 I 𝑥 ∧ 𝐴𝑅𝑥) ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
12	5, 6, 11	3bitri 299	. . . 4 ⊢ (𝐴(𝑅 ∩ I )𝑥 ↔ (𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
13	12	exbii 1842	. . 3 ⊢ (∃𝑥 𝐴(𝑅 ∩ I )𝑥 ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
14	4, 13	bitri 277	. 2 ⊢ (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥))
15		breq2 5061	. . 3 ⊢ (𝑥 = 𝐴 → (𝐴𝑅𝑥 ↔ 𝐴𝑅𝐴))
16	3, 15	ceqsexv 3540	. 2 ⊢ (∃𝑥(𝑥 = 𝐴 ∧ 𝐴𝑅𝑥) ↔ 𝐴𝑅𝐴)
17	2, 14, 16	3bitri 299	1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1531  ∃wex 1774   ∈ wcel 2108  Vcvv 3493   ∩ cin 3933   class class class wbr 5057   I cid 5452  dom cdm 5548   Fix cfix 33289

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-sep 5194  ax-nul 5201  ax-pr 5320

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ral 3141  df-rex 3142  df-rab 3145  df-v 3495  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-nul 4290  df-if 4466  df-sn 4560  df-pr 4562  df-op 4566  df-br 5058  df-opab 5120  df-id 5453  df-xp 5554  df-rel 5555  df-dm 5558  df-fix 33313

This theorem is referenced by:  elfix2  33358  dffix2  33359  fixcnv  33362  ellimits  33364  elfuns  33369  dfrecs2  33404  dfrdg4  33405