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Theorem pet2 38806
Description: Partition-Equivalence Theorem, with general 𝑅. This theorem (together with pet 38807 and pets 38808) is the main result of my investigation into set theory, see the comment of pet 38807. (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 38789 . 2 (( EqvRel ≀ (𝑅 ⋉ ( E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ⋉ ( E ↾ 𝐴)) / ≀ (𝑅 ⋉ ( E ↾ 𝐴))) = 𝐴) → Disj (𝑅 ⋉ ( E ↾ 𝐴)))
21petlem 38768 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 5598  ccnv 5699  dom cdm 5700  cres 5702   / cqs 8762  cxrn 38134  ccoss 38135   EqvRel weqvrel 38152   Disj wdisjALTV 38169
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-eprel 5599  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fo 6579  df-fv 6581  df-1st 8030  df-2nd 8031  df-ec 8765  df-qs 8769  df-xrn 38327  df-coss 38367  df-refrel 38468  df-cnvrefrel 38483  df-symrel 38500  df-trrel 38530  df-eqvrel 38541  df-funALTV 38638  df-disjALTV 38661  df-eldisj 38663
This theorem is referenced by:  pet  38807
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