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Theorem elfix2 32600
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 3414 . 2 (𝐴 Fix 𝑅𝐴 ∈ V)
2 elfix2.1 . . 3 Rel 𝑅
32brrelex1i 5406 . 2 (𝐴𝑅𝐴𝐴 ∈ V)
4 eleq1 2847 . . 3 (𝑥 = 𝐴 → (𝑥 Fix 𝑅𝐴 Fix 𝑅))
5 breq12 4891 . . . 4 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑅𝑥𝐴𝑅𝐴))
65anidms 562 . . 3 (𝑥 = 𝐴 → (𝑥𝑅𝑥𝐴𝑅𝐴))
7 vex 3401 . . . 4 𝑥 ∈ V
87elfix 32599 . . 3 (𝑥 Fix 𝑅𝑥𝑅𝑥)
94, 6, 8vtoclbg 3468 . 2 (𝐴 ∈ V → (𝐴 Fix 𝑅𝐴𝑅𝐴))
101, 3, 9pm5.21nii 370 1 (𝐴 Fix 𝑅𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 198   = wceq 1601  wcel 2107  Vcvv 3398   class class class wbr 4886  Rel wrel 5360   Fix cfix 32531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-sep 5017  ax-nul 5025  ax-pr 5138
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-br 4887  df-opab 4949  df-id 5261  df-xp 5361  df-rel 5362  df-dm 5365  df-fix 32555
This theorem is referenced by: (None)
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