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Mirrors > Home > MPE Home > Th. List > Mathboxes > erimeq | Structured version Visualization version GIF version |
Description: Equivalence relation on its natural domain implies that the class of coelements on the domain is equal to the relation (this is the most convenient form of prter3 38263 and erimeq2 38059). (Contributed by Peter Mazsa, 7-Oct-2021.) (Revised by Peter Mazsa, 29-Dec-2021.) |
Ref | Expression |
---|---|
erimeq | ⊢ (𝑅 ∈ 𝑉 → (𝑅 ErALTV 𝐴 → ∼ 𝐴 = 𝑅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dferALTV2 38049 | . 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) | |
2 | erimeq2 38059 | . 2 ⊢ (𝑅 ∈ 𝑉 → (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ∼ 𝐴 = 𝑅)) | |
3 | 1, 2 | biimtrid 241 | 1 ⊢ (𝑅 ∈ 𝑉 → (𝑅 ErALTV 𝐴 → ∼ 𝐴 = 𝑅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1533 ∈ wcel 2098 dom cdm 5669 / cqs 8701 ∼ ccoels 37555 EqvRel weqvrel 37571 ErALTV werALTV 37580 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-id 5567 df-eprel 5573 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-ec 8704 df-qs 8708 df-coss 37792 df-coels 37793 df-refrel 37893 df-symrel 37925 df-trrel 37955 df-eqvrel 37966 df-dmqs 38020 df-erALTV 38045 |
This theorem is referenced by: partimeq 38190 |
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