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Theorem pet2 38214
Description: Partition-Equivalence Theorem, with general 𝑅. This theorem (together with pet 38215 and pets 38216) is the main result of my investigation into set theory, see the comment of pet 38215. (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 38197 . 2 (( EqvRel ≀ (𝑅 ⋉ ( E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( E ↾ 𝐴)) / ≀ (𝑅 ⋉ ( E ↾ 𝐴))) = 𝐴) → Disj (𝑅 ⋉ ( E ↾ 𝐴)))
21petlem 38176 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 205  wa 395   = wceq 1533   E cep 5570  ccnv 5666  dom cdm 5667  cres 5669   / cqs 8699  cxrn 37536  ccoss 37537   EqvRel weqvrel 37554   Disj wdisjALTV 37571
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418  ax-un 7719
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-eprel 5571  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-iota 6486  df-fun 6536  df-fn 6537  df-f 6538  df-fo 6540  df-fv 6542  df-1st 7969  df-2nd 7970  df-ec 8702  df-qs 8706  df-xrn 37735  df-coss 37775  df-refrel 37876  df-cnvrefrel 37891  df-symrel 37908  df-trrel 37938  df-eqvrel 37949  df-funALTV 38046  df-disjALTV 38069  df-eldisj 38071
This theorem is referenced by:  pet  38215
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