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Mirrors > Home > MPE Home > Th. List > Mathboxes > partim2 | Structured version Visualization version GIF version |
Description: Disjoint relation on its natural domain implies an equivalence relation on the cosets of the relation, on its natural domain, cf. partim 37678. Lemma for petlem 37682. (Contributed by Peter Mazsa, 17-Sep-2021.) |
Ref | Expression |
---|---|
partim2 | ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjim 37651 | . . 3 ⊢ ( Disj 𝑅 → EqvRel ≀ 𝑅) | |
2 | 1 | adantr 482 | . 2 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → EqvRel ≀ 𝑅) |
3 | disjdmqseq 37675 | . . 3 ⊢ ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴)) | |
4 | 3 | biimpa 478 | . 2 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴) |
5 | 2, 4 | jca 513 | 1 ⊢ (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 / ≀ 𝑅) = 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 dom cdm 5677 / cqs 8702 ≀ ccoss 37043 EqvRel weqvrel 37060 Disj wdisjALTV 37077 |
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-ral 3063 df-rex 3072 df-rmo 3377 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-br 5150 df-opab 5212 df-id 5575 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-refrel 37382 df-cnvrefrel 37397 df-symrel 37414 df-trrel 37444 df-eqvrel 37455 df-disjALTV 37575 |
This theorem is referenced by: partim 37678 petlem 37682 |
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