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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 37543 | . 2 ⊢ ( CoMembEr 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (dom ∼ 𝐴 / ∼ 𝐴) = 𝐴)) | |
2 | dfcoeleqvrel 37492 | . . . 4 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ∼ 𝐴) | |
3 | 2 | bicomi 223 | . . 3 ⊢ ( EqvRel ∼ 𝐴 ↔ CoElEqvRel 𝐴) |
4 | dmqscoelseq 37531 | . . 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 205 ∧ wa 397 = wceq 1542 ∪ cuni 4909 dom cdm 5677 / cqs 8702 ∼ ccoels 37044 EqvRel weqvrel 37060 CoElEqvRel wcoeleqvrel 37062 CoMembEr wcomember 37071 |
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-11 2155 ax-12 2172 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
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-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-eprel 5581 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-ec 8705 df-qs 8709 df-coss 37281 df-coels 37282 df-refrel 37382 df-symrel 37414 df-trrel 37444 df-eqvrel 37455 df-coeleqvrel 37457 df-dmqs 37509 df-erALTV 37534 df-comember 37536 |
This theorem is referenced by: mainer 37704 mpet 37709 |
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