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| 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 38668 | . 2 ⊢ ( CoMembEr 𝐴 ↔ ≀ (◡ E ↾ 𝐴) ErALTV 𝐴) | |
| 2 | df-coels 38414 | . . 3 ⊢ ∼ 𝐴 = ≀ (◡ E ↾ 𝐴) | |
| 3 | 2 | erALTVeq1i 38672 | . 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 206 E cep 5582 ◡ccnv 5683 ↾ cres 5686 ≀ ccoss 38183 ∼ ccoels 38184 ErALTV werALTV 38209 CoMembEr wcomember 38211 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ec 8748 df-qs 8752 df-coels 38414 df-refrel 38514 df-symrel 38546 df-trrel 38576 df-eqvrel 38587 df-dmqs 38641 df-erALTV 38666 df-comember 38668 | 
| This theorem is referenced by: dfcomember2 38675 | 
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