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Theorem partim2 38789
Description: Disjoint relation on its natural domain implies an equivalence relation on the cosets of the relation, on its natural domain, cf. partim 38790. Lemma for petlem 38794. (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 38763 . . 3 ( Disj 𝑅 → EqvRel ≀ 𝑅)
21adantr 480 . 2 (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → EqvRel ≀ 𝑅)
3 disjdmqseq 38787 . . 3 ( Disj 𝑅 → ((dom 𝑅 / 𝑅) = 𝐴 ↔ (dom ≀ 𝑅 /𝑅) = 𝐴))
43biimpa 476 . 2 (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (dom ≀ 𝑅 /𝑅) = 𝐴)
52, 4jca 511 1 (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 /𝑅) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1537  dom cdm 5689   / cqs 8743  ccoss 38162   EqvRel weqvrel 38179   Disj wdisjALTV 38196
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-ral 3060  df-rex 3069  df-rmo 3378  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-br 5149  df-opab 5211  df-id 5583  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-refrel 38494  df-cnvrefrel 38509  df-symrel 38526  df-trrel 38556  df-eqvrel 38567  df-disjALTV 38687
This theorem is referenced by:  partim  38790  petlem  38794
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