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Theorem elfix2 32539
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 3429 . 2 (𝐴 Fix 𝑅𝐴 ∈ V)
2 elfix2.1 . . 3 Rel 𝑅
32brrelex1i 5393 . 2 (𝐴𝑅𝐴𝐴 ∈ V)
4 eleq1 2894 . . 3 (𝑥 = 𝐴 → (𝑥 Fix 𝑅𝐴 Fix 𝑅))
5 breq12 4878 . . . 4 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑅𝑥𝐴𝑅𝐴))
65anidms 562 . . 3 (𝑥 = 𝐴 → (𝑥𝑅𝑥𝐴𝑅𝐴))
7 vex 3417 . . . 4 𝑥 ∈ V
87elfix 32538 . . 3 (𝑥 Fix 𝑅𝑥𝑅𝑥)
94, 6, 8vtoclbg 3483 . 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 1656  wcel 2164  Vcvv 3414   class class class wbr 4873  Rel wrel 5347   Fix cfix 32470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5005  ax-nul 5013  ax-pr 5127
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ral 3122  df-rex 3123  df-rab 3126  df-v 3416  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4145  df-if 4307  df-sn 4398  df-pr 4400  df-op 4404  df-br 4874  df-opab 4936  df-id 5250  df-xp 5348  df-rel 5349  df-dm 5352  df-fix 32494
This theorem is referenced by: (None)
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