Theorem fixcnv 36269

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 3461	. . . 4 ⊢ 𝑥 ∈ V
2	1, 1	brcnv 5859	. . 3 ⊢ (𝑥◡𝐴𝑥 ↔ 𝑥𝐴𝑥)
3	1	elfix 36264	. . 3 ⊢ (𝑥 ∈ Fix ◡𝐴 ↔ 𝑥◡𝐴𝑥)
4	1	elfix 36264	. . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥)
5	2, 3, 4	3bitr4ri 307	. 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ Fix ◡𝐴)
6	5	eqriv 2762	1 ⊢ Fix 𝐴 = Fix ◡𝐴

Syntax hints:   = wceq 1563   ∈ wcel 2145   class class class wbr 5105  ^◡ccnv 5651   Fix cfix 36196

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-dm 5662  df-fix 36220

This theorem is referenced by: (None)