Theorem fixcnv 35864

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 3492	. . . 4 ⊢ 𝑥 ∈ V
2	1, 1	brcnv 5902	. . 3 ⊢ (𝑥◡𝐴𝑥 ↔ 𝑥𝐴𝑥)
3	1	elfix 35859	. . 3 ⊢ (𝑥 ∈ Fix ◡𝐴 ↔ 𝑥◡𝐴𝑥)
4	1	elfix 35859	. . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥)
5	2, 3, 4	3bitr4ri 304	. 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ Fix ◡𝐴)
6	5	eqriv 2737	1 ⊢ Fix 𝐴 = Fix ◡𝐴

Syntax hints:   = wceq 1537   ∈ wcel 2108   class class class wbr 5166  ^◡ccnv 5694   Fix cfix 35791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-xp 5701  df-rel 5702  df-cnv 5703  df-dm 5705  df-fix 35815

This theorem is referenced by: (None)