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Theorem partim2 38808
Description: Disjoint relation on its natural domain implies an equivalence relation on the cosets of the relation, on its natural domain, cf. partim 38809. Lemma for petlem 38813. (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 38782 . . 3 ( Disj 𝑅 → EqvRel ≀ 𝑅)
21adantr 480 . 2 (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → EqvRel ≀ 𝑅)
3 disjdmqseq 38806 . . 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 1540  dom cdm 5685   / cqs 8744  ccoss 38182   EqvRel weqvrel 38199   Disj wdisjALTV 38216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751  df-coss 38412  df-refrel 38513  df-cnvrefrel 38528  df-symrel 38545  df-trrel 38575  df-eqvrel 38586  df-disjALTV 38706
This theorem is referenced by:  partim  38809  petlem  38813
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