Theorem elfix2 36045

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 3459	. 2 ⊢ (𝐴 ∈ Fix 𝑅 → 𝐴 ∈ V)
2		elfix2.1	. . 3 ⊢ Rel 𝑅
3	2	brrelex1i 5678	. 2 ⊢ (𝐴𝑅𝐴 → 𝐴 ∈ V)
4		eleq1 2822	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ Fix 𝑅 ↔ 𝐴 ∈ Fix 𝑅))
5		breq12 5101	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
6	5	anidms 566	. . 3 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
7		vex 3442	. . . 4 ⊢ 𝑥 ∈ V
8	7	elfix 36044	. . 3 ⊢ (𝑥 ∈ Fix 𝑅 ↔ 𝑥𝑅𝑥)
9	4, 6, 8	vtoclbg 3512	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴))
10	1, 3, 9	pm5.21nii 378	1 ⊢ (𝐴 ∈ Fix 𝑅 ↔ 𝐴𝑅𝐴)

Syntax hints:   ↔ wb 206   = wceq 1541   ∈ wcel 2113  Vcvv 3438   class class class wbr 5096  Rel wrel 5627   Fix cfix 35976

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-br 5097  df-opab 5159  df-id 5517  df-xp 5628  df-rel 5629  df-dm 5632  df-fix 36000

This theorem is referenced by: (None)