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Theorem elfix 33352
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 33308 . . 3 Fix 𝑅 = dom (𝑅 ∩ I )
21eleq2i 2902 . 2 (𝐴 Fix 𝑅𝐴 ∈ dom (𝑅 ∩ I ))
3 elfix.1 . . . 4 𝐴 ∈ V
43eldm 5762 . . 3 (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥)
5 brin 5109 . . . . 5 (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥𝐴 I 𝑥))
6 ancom 463 . . . . 5 ((𝐴𝑅𝑥𝐴 I 𝑥) ↔ (𝐴 I 𝑥𝐴𝑅𝑥))
7 vex 3496 . . . . . . . 8 𝑥 ∈ V
87ideq 5716 . . . . . . 7 (𝐴 I 𝑥𝐴 = 𝑥)
9 eqcom 2826 . . . . . . 7 (𝐴 = 𝑥𝑥 = 𝐴)
108, 9bitri 277 . . . . . 6 (𝐴 I 𝑥𝑥 = 𝐴)
1110anbi1i 625 . . . . 5 ((𝐴 I 𝑥𝐴𝑅𝑥) ↔ (𝑥 = 𝐴𝐴𝑅𝑥))
125, 6, 113bitri 299 . . . 4 (𝐴(𝑅 ∩ I )𝑥 ↔ (𝑥 = 𝐴𝐴𝑅𝑥))
1312exbii 1841 . . 3 (∃𝑥 𝐴(𝑅 ∩ I )𝑥 ↔ ∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥))
144, 13bitri 277 . 2 (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥))
15 breq2 5061 . . 3 (𝑥 = 𝐴 → (𝐴𝑅𝑥𝐴𝑅𝐴))
163, 15ceqsexv 3540 . 2 (∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥) ↔ 𝐴𝑅𝐴)
172, 14, 163bitri 299 1 (𝐴 Fix 𝑅𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 208  wa 398   = wceq 1530  wex 1773  wcel 2107  Vcvv 3493  cin 3933   class class class wbr 5057   I cid 5452  dom cdm 5548   Fix cfix 33284
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  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 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  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 33308
This theorem is referenced by:  elfix2  33353  dffix2  33354  fixcnv  33357  ellimits  33359  elfuns  33364  dfrecs2  33399  dfrdg4  33400
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