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Theorem elfix2 35631
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 3480 . 2 (𝐴 Fix 𝑅𝐴 ∈ V)
2 elfix2.1 . . 3 Rel 𝑅
32brrelex1i 5734 . 2 (𝐴𝑅𝐴𝐴 ∈ V)
4 eleq1 2813 . . 3 (𝑥 = 𝐴 → (𝑥 Fix 𝑅𝐴 Fix 𝑅))
5 breq12 5154 . . . 4 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑅𝑥𝐴𝑅𝐴))
65anidms 565 . . 3 (𝑥 = 𝐴 → (𝑥𝑅𝑥𝐴𝑅𝐴))
7 vex 3465 . . . 4 𝑥 ∈ V
87elfix 35630 . . 3 (𝑥 Fix 𝑅𝑥𝑅𝑥)
94, 6, 8vtoclbg 3535 . 2 (𝐴 ∈ V → (𝐴 Fix 𝑅𝐴𝑅𝐴))
101, 3, 9pm5.21nii 377 1 (𝐴 Fix 𝑅𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wcel 2098  Vcvv 3461   class class class wbr 5149  Rel wrel 5683   Fix cfix 35562
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5150  df-opab 5212  df-id 5576  df-xp 5684  df-rel 5685  df-dm 5688  df-fix 35586
This theorem is referenced by: (None)
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