Theorem fixcnv 35889

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 3448	. . . 4 ⊢ 𝑥 ∈ V
2	1, 1	brcnv 5836	. . 3 ⊢ (𝑥◡𝐴𝑥 ↔ 𝑥𝐴𝑥)
3	1	elfix 35884	. . 3 ⊢ (𝑥 ∈ Fix ◡𝐴 ↔ 𝑥◡𝐴𝑥)
4	1	elfix 35884	. . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥)
5	2, 3, 4	3bitr4ri 304	. 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ Fix ◡𝐴)
6	5	eqriv 2726	1 ⊢ Fix 𝐴 = Fix ◡𝐴

Syntax hints:   = wceq 1540   ∈ wcel 2109   class class class wbr 5102  ^◡ccnv 5630   Fix cfix 35816

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-br 5103  df-opab 5165  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-dm 5641  df-fix 35840

This theorem is referenced by: (None)