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Theorem elfix 32549
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 32505 . . 3 Fix 𝑅 = dom (𝑅 ∩ I )
21eleq2i 2898 . 2 (𝐴 Fix 𝑅𝐴 ∈ dom (𝑅 ∩ I ))
3 elfix.1 . . . 4 𝐴 ∈ V
43eldm 5553 . . 3 (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥)
5 brin 4925 . . . . 5 (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥𝐴 I 𝑥))
6 ancom 454 . . . . 5 ((𝐴𝑅𝑥𝐴 I 𝑥) ↔ (𝐴 I 𝑥𝐴𝑅𝑥))
7 vex 3417 . . . . . . . 8 𝑥 ∈ V
87ideq 5507 . . . . . . 7 (𝐴 I 𝑥𝐴 = 𝑥)
9 eqcom 2832 . . . . . . 7 (𝐴 = 𝑥𝑥 = 𝐴)
108, 9bitri 267 . . . . . 6 (𝐴 I 𝑥𝑥 = 𝐴)
1110anbi1i 619 . . . . 5 ((𝐴 I 𝑥𝐴𝑅𝑥) ↔ (𝑥 = 𝐴𝐴𝑅𝑥))
125, 6, 113bitri 289 . . . 4 (𝐴(𝑅 ∩ I )𝑥 ↔ (𝑥 = 𝐴𝐴𝑅𝑥))
1312exbii 1949 . . 3 (∃𝑥 𝐴(𝑅 ∩ I )𝑥 ↔ ∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥))
144, 13bitri 267 . 2 (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥))
15 breq2 4877 . . 3 (𝑥 = 𝐴 → (𝐴𝑅𝑥𝐴𝑅𝐴))
163, 15ceqsexv 3459 . 2 (∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥) ↔ 𝐴𝑅𝐴)
172, 14, 163bitri 289 1 (𝐴 Fix 𝑅𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 386   = wceq 1658  wex 1880  wcel 2166  Vcvv 3414  cin 3797   class class class wbr 4873   I cid 5249  dom cdm 5342   Fix cfix 32481
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2391  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-dm 5352  df-fix 32505
This theorem is referenced by:  elfix2  32550  dffix2  32551  fixcnv  32554  ellimits  32556  elfuns  32561  dfrecs2  32596  dfrdg4  32597
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