![]() |
Mathbox for Scott Fenton |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > fixcnv | Structured version Visualization version GIF version |
Description: The fixpoints of a class are the same as those of its converse. (Contributed by Scott Fenton, 16-Apr-2012.) |
Ref | Expression |
---|---|
fixcnv | ⊢ Fix 𝐴 = Fix ◡𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3481 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1, 1 | brcnv 5895 | . . 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 2731 | 1 ⊢ Fix 𝐴 = Fix ◡𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ◡ccnv 5687 Fix cfix 35816 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-dm 5698 df-fix 35840 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |