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Theorem elfix 32347
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 32303 . . 3 Fix 𝑅 = dom (𝑅 ∩ I )
21eleq2i 2842 . 2 (𝐴 Fix 𝑅𝐴 ∈ dom (𝑅 ∩ I ))
3 elfix.1 . . . 4 𝐴 ∈ V
43eldm 5458 . . 3 (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥)
5 brin 4839 . . . . 5 (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥𝐴 I 𝑥))
6 ancom 448 . . . . 5 ((𝐴𝑅𝑥𝐴 I 𝑥) ↔ (𝐴 I 𝑥𝐴𝑅𝑥))
7 vex 3354 . . . . . . . 8 𝑥 ∈ V
87ideq 5412 . . . . . . 7 (𝐴 I 𝑥𝐴 = 𝑥)
9 eqcom 2778 . . . . . . 7 (𝐴 = 𝑥𝑥 = 𝐴)
108, 9bitri 264 . . . . . 6 (𝐴 I 𝑥𝑥 = 𝐴)
1110anbi1i 610 . . . . 5 ((𝐴 I 𝑥𝐴𝑅𝑥) ↔ (𝑥 = 𝐴𝐴𝑅𝑥))
125, 6, 113bitri 286 . . . 4 (𝐴(𝑅 ∩ I )𝑥 ↔ (𝑥 = 𝐴𝐴𝑅𝑥))
1312exbii 1924 . . 3 (∃𝑥 𝐴(𝑅 ∩ I )𝑥 ↔ ∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥))
144, 13bitri 264 . 2 (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥))
15 breq2 4791 . . 3 (𝑥 = 𝐴 → (𝐴𝑅𝑥𝐴𝑅𝐴))
163, 15ceqsexv 3394 . 2 (∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥) ↔ 𝐴𝑅𝐴)
172, 14, 163bitri 286 1 (𝐴 Fix 𝑅𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 382   = wceq 1631  wex 1852  wcel 2145  Vcvv 3351  cin 3722   class class class wbr 4787   I cid 5157  dom cdm 5250   Fix cfix 32279
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-id 5158  df-xp 5256  df-rel 5257  df-dm 5260  df-fix 32303
This theorem is referenced by:  elfix2  32348  dffix2  32349  fixcnv  32352  ellimits  32354  elfuns  32359  dfrecs2  32394  dfrdg4  32395
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