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Theorem elfix2 33479
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 3462 . 2 (𝐴 Fix 𝑅𝐴 ∈ V)
2 elfix2.1 . . 3 Rel 𝑅
32brrelex1i 5576 . 2 (𝐴𝑅𝐴𝐴 ∈ V)
4 eleq1 2880 . . 3 (𝑥 = 𝐴 → (𝑥 Fix 𝑅𝐴 Fix 𝑅))
5 breq12 5038 . . . 4 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑅𝑥𝐴𝑅𝐴))
65anidms 570 . . 3 (𝑥 = 𝐴 → (𝑥𝑅𝑥𝐴𝑅𝐴))
7 vex 3447 . . . 4 𝑥 ∈ V
87elfix 33478 . . 3 (𝑥 Fix 𝑅𝑥𝑅𝑥)
94, 6, 8vtoclbg 3520 . 2 (𝐴 ∈ V → (𝐴 Fix 𝑅𝐴𝑅𝐴))
101, 3, 9pm5.21nii 383 1 (𝐴 Fix 𝑅𝐴𝑅𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 209   = wceq 1538  wcel 2112  Vcvv 3444   class class class wbr 5033  Rel wrel 5528   Fix cfix 33410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-id 5428  df-xp 5529  df-rel 5530  df-dm 5533  df-fix 33434
This theorem is referenced by: (None)
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