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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 38655 | . 2 ⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴)) | |
2 | dfcoeleqvrel 38604 | . . . 4 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴) | |
3 | 2 | bicomi 224 | . . 3 ⊢ ( EqvRel ∼ 𝐴 ↔ CoElEqvRel 𝐴) |
4 | dmqscoelseq 38643 | . . 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 1537 ∪ cuni 4912 dom cdm 5689 / cqs 8743 ∼ ccoels 38163 EqvRel weqvrel 38179 CoElEqvRel wcoeleqvrel 38181 CoMembEr wcomember 38190 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-eprel 5589 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ec 8746 df-qs 8750 df-coss 38393 df-coels 38394 df-refrel 38494 df-symrel 38526 df-trrel 38556 df-eqvrel 38567 df-coeleqvrel 38569 df-dmqs 38621 df-erALTV 38646 df-comember 38648 |
This theorem is referenced by: mainer 38816 mpet 38821 |
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