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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfcomember2 | Structured version Visualization version GIF version |
Description: Alternate definition of the comember equivalence relation. (Contributed by Peter Mazsa, 25-Sep-2021.) |
Ref | Expression |
---|---|
dfcomember2 | ⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfcomember 38653 | . 2 ⊢ ( CoMembEr 𝐴 ↔ ∼ 𝐴 ErALTV 𝐴) | |
2 | dferALTV2 38649 | . 2 ⊢ ( ∼ 𝐴 ErALTV 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴)) | |
3 | 1, 2 | bitri 275 | 1 ⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1536 dom cdm 5688 / cqs 8742 ∼ ccoels 38162 EqvRel weqvrel 38178 ErALTV werALTV 38187 CoMembEr wcomember 38189 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-br 5148 df-opab 5210 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ec 8745 df-qs 8749 df-coels 38393 df-refrel 38493 df-symrel 38525 df-trrel 38555 df-eqvrel 38566 df-dmqs 38620 df-erALTV 38645 df-comember 38647 |
This theorem is referenced by: dfcomember3 38655 |
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