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 38245
Description: The shortest form of a partition-equivalence theorem with intersection and general 𝑅. Cf. br1cossincnvepres 37846. Cf. pet 38247. (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 38244 . 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 38165 . 2 ((𝑅 ∩ ( E ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ∩ ( E ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( E ↾ 𝐴)) / (𝑅 ∩ ( E ↾ 𝐴))) = 𝐴))
3 dferALTV2 38064 . 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 205  wa 395   = wceq 1534  cin 3943   E cep 5575  ccnv 5671  dom cdm 5672  cres 5674   / cqs 8715  ccoss 37570   EqvRel weqvrel 37587   ErALTV werALTV 37596   Disj wdisjALTV 37604   Part wpart 37609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rmo 3371  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-id 5570  df-eprel 5576  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-ec 8718  df-qs 8722  df-coss 37807  df-refrel 37908  df-cnvrefrel 37923  df-symrel 37940  df-trrel 37970  df-eqvrel 37981  df-dmqs 38035  df-erALTV 38060  df-funALTV 38078  df-disjALTV 38101  df-eldisj 38103  df-part 38162
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator