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Theorem elfix2 31680
Description: Alternative membership in the fixpoint of a class. (Contributed by Scott Fenton, 11-Apr-2012.)
Hypothesis
Ref Expression
elfix2.1 Rel 𝑅
Assertion
Ref Expression
elfix2 (𝐴 Fix 𝑅𝐴𝑅𝐴)

Proof of Theorem elfix2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elex 3201 . 2 (𝐴 Fix 𝑅𝐴 ∈ V)
2 elfix2.1 . . 3 Rel 𝑅
32brrelexi 5123 . 2 (𝐴𝑅𝐴𝐴 ∈ V)
4 eleq1 2686 . . 3 (𝑥 = 𝐴 → (𝑥 Fix 𝑅𝐴 Fix 𝑅))
5 breq12 4623 . . . 4 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑅𝑥𝐴𝑅𝐴))
65anidms 676 . . 3 (𝑥 = 𝐴 → (𝑥𝑅𝑥𝐴𝑅𝐴))
7 vex 3192 . . . 4 𝑥 ∈ V
87elfix 31679 . . 3 (𝑥 Fix 𝑅𝑥𝑅𝑥)
94, 6, 8vtoclbg 3256 . 2 (𝐴 ∈ V → (𝐴 Fix 𝑅𝐴𝑅𝐴))
101, 3, 9pm5.21nii 368 1 (𝐴 Fix 𝑅𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196   = wceq 1480  wcel 1987  Vcvv 3189   class class class wbr 4618  Rel wrel 5084   Fix cfix 31610
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 31634
This theorem is referenced by: (None)
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