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Theorem partim2 37763
Description: Disjoint relation on its natural domain implies an equivalence relation on the cosets of the relation, on its natural domain, cf. partim 37764. Lemma for petlem 37768. (Contributed by Peter Mazsa, 17-Sep-2021.)
Assertion
Ref Expression
partim2 (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 /𝑅) = 𝐴))

Proof of Theorem partim2
StepHypRef Expression
1 disjim 37737 . . 3 ( Disj 𝑅 → EqvRel ≀ 𝑅)
21adantr 481 . 2 (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → EqvRel ≀ 𝑅)
3 disjdmqseq 37761 . . 3 ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 /𝑅) = 𝐴))
43biimpa 477 . 2 (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (dom ≀ 𝑅 /𝑅) = 𝐴)
52, 4jca 512 1 (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 /𝑅) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1541  dom cdm 5676   / cqs 8704  ccoss 37129   EqvRel weqvrel 37146   Disj wdisjALTV 37163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rmo 3376  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8707  df-qs 8711  df-coss 37367  df-refrel 37468  df-cnvrefrel 37483  df-symrel 37500  df-trrel 37530  df-eqvrel 37541  df-disjALTV 37661
This theorem is referenced by:  partim  37764  petlem  37768
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