Theorem fixcnv 33482

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 3444	. . . 4 ⊢ 𝑥 ∈ V
2	1, 1	brcnv 5717	. . 3 ⊢ (𝑥◡𝐴𝑥 ↔ 𝑥𝐴𝑥)
3	1	elfix 33477	. . 3 ⊢ (𝑥 ∈ Fix ◡𝐴 ↔ 𝑥◡𝐴𝑥)
4	1	elfix 33477	. . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥)
5	2, 3, 4	3bitr4ri 307	. 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ Fix ◡𝐴)
6	5	eqriv 2795	1 ⊢ Fix 𝐴 = Fix ◡𝐴

Syntax hints:   = wceq 1538   ∈ wcel 2111   class class class wbr 5030  ^◡ccnv 5518   Fix cfix 33409

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-dm 5529  df-fix 33433

This theorem is referenced by: (None)