Theorem elfix2 36118

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 𝑅 ↔ 𝐴𝑅𝐴)

Proof of Theorem elfix2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3463	. 2 ⊢ (𝐴 ∈ Fix 𝑅 → 𝐴 ∈ V)
2		elfix2.1	. . 3 ⊢ Rel 𝑅
3	2	brrelex1i 5688	. 2 ⊢ (𝐴𝑅𝐴 → 𝐴 ∈ V)
4		eleq1 2825	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Fix 𝑅 ↔ 𝐴 ∈ Fix 𝑅))
5		breq12 5105	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
6	5	anidms 566	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
7		vex 3446	. . . 4 ⊢ 𝑥 ∈ V
8	7	elfix 36117	. . 3 ⊢ (𝑥 ∈ Fix 𝑅 ↔ 𝑥𝑅𝑥)
9	4, 6, 8	vtoclbg 3516	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴))
10	1, 3, 9	pm5.21nii 378	1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)

Syntax hints:   ↔ wb 206   = wceq 1542   ∈ wcel 2114  Vcvv 3442   class class class wbr 5100  Rel wrel 5637   Fix cfix 36049

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-id 5527  df-xp 5638  df-rel 5639  df-dm 5642  df-fix 36073

This theorem is referenced by: (None)