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Theorem partim 38790
Description: Partition implies equivalence relation by the cosets of the relation on its natural domain, cf. partim2 38789. (Contributed by Peter Mazsa, 17-Sep-2021.)
Assertion
Ref Expression
partim (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴)

Proof of Theorem partim
StepHypRef Expression
1 partim2 38789 . 2 (( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 /𝑅) = 𝐴))
2 dfpart2 38751 . 2 (𝑅 Part 𝐴 ↔ ( Disj 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
3 dferALTV2 38650 . 2 ( ≀ 𝑅 ErALTV 𝐴 ↔ ( EqvRel ≀ 𝑅 ∧ (dom ≀ 𝑅 /𝑅) = 𝐴))
41, 2, 33imtr4i 292 1 (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 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   ErALTV werALTV 38188   Disj wdisjALTV 38196   Part wpart 38201
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-dmqs 38621  df-erALTV 38646  df-disjALTV 38687  df-part 38748
This theorem is referenced by:  partimeq  38791  partimcomember  38817
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