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| Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcomember3 | Structured version Visualization version GIF version | ||
| Description: Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 26-Sep-2021.) (Revised by Peter Mazsa, 17-Jul-2023.) | 
| Ref | Expression | 
|---|---|
| dfcomember3 | ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | dfcomember2 38675 | . 2 ⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴)) | |
| 2 | dfcoeleqvrel 38624 | . . . 4 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴) | |
| 3 | 2 | bicomi 224 | . . 3 ⊢ ( EqvRel ∼ 𝐴 ↔ CoElEqvRel 𝐴) | 
| 4 | dmqscoelseq 38663 | . . 3 ⊢ ((dom ∼ 𝐴 / ∼ 𝐴) = 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴) | |
| 5 | 3, 4 | anbi12i 628 | . 2 ⊢ (( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | 
| 6 | 1, 5 | bitri 275 | 1 ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∪ cuni 4906 dom cdm 5684 / cqs 8745 ∼ ccoels 38184 EqvRel weqvrel 38200 CoElEqvRel wcoeleqvrel 38202 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-10 2140 ax-11 2156 ax-12 2176 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-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 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-uni 4907 df-br 5143 df-opab 5205 df-eprel 5583 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-coss 38413 df-coels 38414 df-refrel 38514 df-symrel 38546 df-trrel 38576 df-eqvrel 38587 df-coeleqvrel 38589 df-dmqs 38641 df-erALTV 38666 df-comember 38668 | 
| This theorem is referenced by: mainer 38836 mpet 38841 | 
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