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Theorem pet2 38832
Description: Partition-Equivalence Theorem, with general 𝑅. This theorem (together with pet 38833 and pets 38834) is the main result of my investigation into set theory, see the comment of pet 38833. (Contributed by Peter Mazsa, 24-May-2021.) (Revised by Peter Mazsa, 23-Sep-2021.)
Assertion
Ref Expression
pet2 (( Disj (𝑅 ⋉ ( E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ ( E ↾ 𝐴)) / (𝑅 ⋉ ( E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ ( E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( E ↾ 𝐴)) / ≀ (𝑅 ⋉ ( E ↾ 𝐴))) = 𝐴))

Proof of Theorem pet2
StepHypRef Expression
1 eqvrelqseqdisj5 38815 . 2 (( EqvRel ≀ (𝑅 ⋉ ( E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( E ↾ 𝐴)) / ≀ (𝑅 ⋉ ( E ↾ 𝐴))) = 𝐴) → Disj (𝑅 ⋉ ( E ↾ 𝐴)))
21petlem 38794 1 (( Disj (𝑅 ⋉ ( E ↾ 𝐴)) ∧ (dom (𝑅 ⋉ ( E ↾ 𝐴)) / (𝑅 ⋉ ( E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ⋉ ( E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( E ↾ 𝐴)) / ≀ (𝑅 ⋉ ( E ↾ 𝐴))) = 𝐴))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537   E cep 5588  ccnv 5688  dom cdm 5689  cres 5691   / cqs 8743  cxrn 38161  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  ax-un 7754
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-ne 2939  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-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-eprel 5589  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-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-ec 8746  df-qs 8750  df-xrn 38353  df-coss 38393  df-refrel 38494  df-cnvrefrel 38509  df-symrel 38526  df-trrel 38556  df-eqvrel 38567  df-funALTV 38664  df-disjALTV 38687  df-eldisj 38689
This theorem is referenced by:  pet  38833
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