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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 3498 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1, 1 | brcnv 5748 | . . 3 ⊢ (𝑥◡𝐴𝑥 ↔ 𝑥𝐴𝑥) |
3 | 1 | elfix 33359 | . . 3 ⊢ (𝑥 ∈ Fix ◡𝐴 ↔ 𝑥◡𝐴𝑥) |
4 | 1 | elfix 33359 | . . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥) |
5 | 2, 3, 4 | 3bitr4ri 306 | . 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ Fix ◡𝐴) |
6 | 5 | eqriv 2818 | 1 ⊢ Fix 𝐴 = Fix ◡𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∈ wcel 2110 class class class wbr 5059 ◡ccnv 5549 Fix cfix 33291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-id 5455 df-xp 5556 df-rel 5557 df-cnv 5558 df-dm 5560 df-fix 33315 |
This theorem is referenced by: (None) |
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