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Theorem elfix 35884
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 𝑅𝐴𝑅𝐴)

Proof of Theorem elfix
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 df-fix 35840 . . 3 Fix 𝑅 = dom (𝑅 ∩ I )
21eleq2i 2830 . 2 (𝐴 Fix 𝑅𝐴 ∈ dom (𝑅 ∩ I ))
3 elfix.1 . . . 4 𝐴 ∈ V
43eldm 5913 . . 3 (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥)
5 brin 5199 . . . . 5 (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥𝐴 I 𝑥))
6 ancom 460 . . . . 5 ((𝐴𝑅𝑥𝐴 I 𝑥) ↔ (𝐴 I 𝑥𝐴𝑅𝑥))
7 vex 3481 . . . . . . . 8 𝑥 ∈ V
87ideq 5865 . . . . . . 7 (𝐴 I 𝑥𝐴 = 𝑥)
9 eqcom 2741 . . . . . . 7 (𝐴 = 𝑥𝑥 = 𝐴)
108, 9bitri 275 . . . . . 6 (𝐴 I 𝑥𝑥 = 𝐴)
1110anbi1i 624 . . . . 5 ((𝐴 I 𝑥𝐴𝑅𝑥) ↔ (𝑥 = 𝐴𝐴𝑅𝑥))
125, 6, 113bitri 297 . . . 4 (𝐴(𝑅 ∩ I )𝑥 ↔ (𝑥 = 𝐴𝐴𝑅𝑥))
1312exbii 1844 . . 3 (∃𝑥 𝐴(𝑅 ∩ I )𝑥 ↔ ∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥))
144, 13bitri 275 . 2 (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥))
15 breq2 5151 . . 3 (𝑥 = 𝐴 → (𝐴𝑅𝑥𝐴𝑅𝐴))
163, 15ceqsexv 3529 . 2 (∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥) ↔ 𝐴𝑅𝐴)
172, 14, 163bitri 297 1 (𝐴 Fix 𝑅𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1536  wex 1775  wcel 2105  Vcvv 3477  cin 3961   class class class wbr 5147   I cid 5581  dom cdm 5688   Fix cfix 35816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-id 5582  df-xp 5694  df-rel 5695  df-dm 5698  df-fix 35840
This theorem is referenced by:  elfix2  35885  dffix2  35886  fixcnv  35889  ellimits  35891  elfuns  35896  dfrecs2  35931  dfrdg4  35932
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