Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  petincnvepres Structured version   Visualization version   GIF version

Theorem petincnvepres 38831
Description: The shortest form of a partition-equivalence theorem with intersection and general 𝑅. Cf. br1cossincnvepres 38432. Cf. pet 38833. (Contributed by Peter Mazsa, 23-Sep-2021.)
Assertion
Ref Expression
petincnvepres ((𝑅 ∩ ( E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ ( E ↾ 𝐴)) ErALTV 𝐴)

Proof of Theorem petincnvepres
StepHypRef Expression
1 petincnvepres2 38830 . 2 (( Disj (𝑅 ∩ ( E ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( E ↾ 𝐴)) / (𝑅 ∩ ( E ↾ 𝐴))) = 𝐴) ↔ ( EqvRel ≀ (𝑅 ∩ ( E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ ( E ↾ 𝐴)) / ≀ (𝑅 ∩ ( E ↾ 𝐴))) = 𝐴))
2 dfpart2 38751 . 2 ((𝑅 ∩ ( E ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ∩ ( E ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( E ↾ 𝐴)) / (𝑅 ∩ ( E ↾ 𝐴))) = 𝐴))
3 dferALTV2 38650 . 2 ( ≀ (𝑅 ∩ ( E ↾ 𝐴)) ErALTV 𝐴 ↔ ( EqvRel ≀ (𝑅 ∩ ( E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ ( E ↾ 𝐴)) / ≀ (𝑅 ∩ ( E ↾ 𝐴))) = 𝐴))
41, 2, 33bitr4i 303 1 ((𝑅 ∩ ( E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ ( E ↾ 𝐴)) ErALTV 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1537  cin 3962   E cep 5588  ccnv 5688  dom cdm 5689  cres 5691   / 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-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-br 5149  df-opab 5211  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-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-funALTV 38664  df-disjALTV 38687  df-eldisj 38689  df-part 38748
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator