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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 3417 | . . . 4 ⊢ 𝑥 ∈ V | |
2 | 1, 1 | brcnv 5541 | . . 3 ⊢ (𝑥◡𝐴𝑥 ↔ 𝑥𝐴𝑥) |
3 | 1 | elfix 32544 | . . 3 ⊢ (𝑥 ∈ Fix ◡𝐴 ↔ 𝑥◡𝐴𝑥) |
4 | 1 | elfix 32544 | . . 3 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥𝐴𝑥) |
5 | 2, 3, 4 | 3bitr4ri 296 | . 2 ⊢ (𝑥 ∈ Fix 𝐴 ↔ 𝑥 ∈ Fix ◡𝐴) |
6 | 5 | eqriv 2822 | 1 ⊢ Fix 𝐴 = Fix ◡𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1656 ∈ wcel 2164 class class class wbr 4875 ◡ccnv 5345 Fix cfix 32476 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-op 4406 df-br 4876 df-opab 4938 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-dm 5356 df-fix 32500 |
This theorem is referenced by: (None) |
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