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Theorem elfix 31687
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 31642 . . 3 Fix 𝑅 = dom (𝑅 ∩ I )
21eleq2i 2690 . 2 (𝐴 Fix 𝑅𝐴 ∈ dom (𝑅 ∩ I ))
3 elfix.1 . . . 4 𝐴 ∈ V
43eldm 5286 . . 3 (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥 𝐴(𝑅 ∩ I )𝑥)
5 brin 4669 . . . . 5 (𝐴(𝑅 ∩ I )𝑥 ↔ (𝐴𝑅𝑥𝐴 I 𝑥))
6 ancom 466 . . . . 5 ((𝐴𝑅𝑥𝐴 I 𝑥) ↔ (𝐴 I 𝑥𝐴𝑅𝑥))
7 vex 3192 . . . . . . . 8 𝑥 ∈ V
87ideq 5239 . . . . . . 7 (𝐴 I 𝑥𝐴 = 𝑥)
9 eqcom 2628 . . . . . . 7 (𝐴 = 𝑥𝑥 = 𝐴)
108, 9bitri 264 . . . . . 6 (𝐴 I 𝑥𝑥 = 𝐴)
1110anbi1i 730 . . . . 5 ((𝐴 I 𝑥𝐴𝑅𝑥) ↔ (𝑥 = 𝐴𝐴𝑅𝑥))
125, 6, 113bitri 286 . . . 4 (𝐴(𝑅 ∩ I )𝑥 ↔ (𝑥 = 𝐴𝐴𝑅𝑥))
1312exbii 1771 . . 3 (∃𝑥 𝐴(𝑅 ∩ I )𝑥 ↔ ∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥))
144, 13bitri 264 . 2 (𝐴 ∈ dom (𝑅 ∩ I ) ↔ ∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥))
15 breq2 4622 . . 3 (𝑥 = 𝐴 → (𝐴𝑅𝑥𝐴𝑅𝐴))
163, 15ceqsexv 3231 . 2 (∃𝑥(𝑥 = 𝐴𝐴𝑅𝑥) ↔ 𝐴𝑅𝐴)
172, 14, 163bitri 286 1 (𝐴 Fix 𝑅𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384   = wceq 1480  wex 1701  wcel 1987  Vcvv 3189  cin 3558   class class class wbr 4618   I cid 4989  dom cdm 5079   Fix cfix 31618
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pr 4872
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-dm 5089  df-fix 31642
This theorem is referenced by:  elfix2  31688  dffix2  31689  fixcnv  31692  ellimits  31694  elfuns  31699  dfrecs2  31734  dfrdg4  31735
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