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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcomember | Structured version Visualization version GIF version |
Description: Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 28-Nov-2022.) |
Ref | Expression |
---|---|
dfcomember | ⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-comember 37474 | . 2 ⊢ ( CoMembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴) | |
2 | df-coels 37220 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
3 | 2 | erALTVeq1i 37478 | . 2 ⊢ ( ∼ 𝐴 ErALTV 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴) |
4 | 1, 3 | bitr4i 278 | 1 ⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 E cep 5578 ◡ccnv 5674 ↾ cres 5677 ≀ ccoss 36981 ∼ ccoels 36982 ErALTV werALTV 37007 CoMembEr wcomember 37009 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ec 8701 df-qs 8705 df-coels 37220 df-refrel 37320 df-symrel 37352 df-trrel 37382 df-eqvrel 37393 df-dmqs 37447 df-erALTV 37472 df-comember 37474 |
This theorem is referenced by: dfcomember2 37481 |
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