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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 3495 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1, 1 | brcnv 5746 | . . 3 ⊢ (𝑥◡𝐴𝑥 ↔ 𝑥𝐴𝑥) |
3 | 1 | elfix 33261 | . . 3 ⊢ (𝑥 ∈ Fix ◡𝐴 ↔ 𝑥◡𝐴𝑥) |
4 | 1 | elfix 33261 | . . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥) |
5 | 2, 3, 4 | 3bitr4ri 305 | . 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ Fix ◡𝐴) |
6 | 5 | eqriv 2815 | 1 ⊢ Fix 𝐴 = Fix ◡𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1528 ∈ wcel 2105 class class class wbr 5057 ◡ccnv 5547 Fix cfix 33193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-dm 5558 df-fix 33217 |
This theorem is referenced by: (None) |
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