Theorem fixcnv 35734

Description: The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.)

Assertion

Ref	Expression
fixcnv	⊢ Fix 𝐴 = Fix ◡𝐴

Proof of Theorem fixcnv

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 3466	. . . 4 ⊢ 𝑥 ∈ V
2	1, 1	brcnv 5891	. . 3 ⊢ (𝑥◡𝐴𝑥 ↔ 𝑥𝐴𝑥)
3	1	elfix 35729	. . 3 ⊢ (𝑥 ∈ Fix ◡𝐴 ↔ 𝑥◡𝐴𝑥)
4	1	elfix 35729	. . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥)
5	2, 3, 4	3bitr4ri 303	. 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ Fix ◡𝐴)
6	5	eqriv 2723	1 ⊢ Fix 𝐴 = Fix ◡𝐴

Syntax hints:   = wceq 1534   ∈ wcel 2099   class class class wbr 5155  ^◡ccnv 5683   Fix cfix 35661

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5306  ax-nul 5313  ax-pr 5435

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5156  df-opab 5218  df-id 5582  df-xp 5690  df-rel 5691  df-cnv 5692  df-dm 5694  df-fix 35685

This theorem is referenced by: (None)