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Theorem partimeq 38810
Description: Partition implies that the class of coelements on the natural domain is equal to the class of cosets of the relation, cf. erimeq 38680. (Contributed by Peter Mazsa, 25-Dec-2024.)
Assertion
Ref Expression
partimeq (𝑅𝑉 → (𝑅 Part 𝐴 → ∼ 𝐴 = ≀ 𝑅))

Proof of Theorem partimeq
StepHypRef Expression
1 cossex 38420 . 2 (𝑅𝑉 → ≀ 𝑅 ∈ V)
2 partim 38809 . 2 (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)
3 erimeq 38680 . 2 ( ≀ 𝑅 ∈ V → ( ≀ 𝑅 ErALTV 𝐴 → ∼ 𝐴 = ≀ 𝑅))
41, 2, 3syl2im 40 1 (𝑅𝑉 → (𝑅 Part 𝐴 → ∼ 𝐴 = ≀ 𝑅))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  ccoss 38182  ccoels 38183   ErALTV werALTV 38208   Part wpart 38221
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-pow 5365  ax-pr 5432  ax-un 7755
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-ne 2941  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-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-id 5578  df-eprel 5584  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-coels 38413  df-refrel 38513  df-cnvrefrel 38528  df-symrel 38545  df-trrel 38575  df-eqvrel 38586  df-dmqs 38640  df-erALTV 38665  df-disjALTV 38706  df-part 38767
This theorem is referenced by: (None)
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