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 38373
Description: The shortest form of a partition-equivalence theorem with intersection and general 𝑅. Cf. br1cossincnvepres 37974. Cf. pet 38375. (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 38372 . 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 38293 . 2 ((𝑅 ∩ ( E ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ∩ ( E ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( E ↾ 𝐴)) / (𝑅 ∩ ( E ↾ 𝐴))) = 𝐴))
3 dferALTV2 38192 . 2 ( ≀ (𝑅 ∩ ( E ↾ 𝐴)) ErALTV 𝐴 ↔ ( EqvRel ≀ (𝑅 ∩ ( E ↾ 𝐴)) ∧ (dom ≀ (𝑅 ∩ ( E ↾ 𝐴)) / ≀ (𝑅 ∩ ( E ↾ 𝐴))) = 𝐴))
41, 2, 33bitr4i 302 1 ((𝑅 ∩ ( E ↾ 𝐴)) Part 𝐴 ↔ ≀ (𝑅 ∩ ( E ↾ 𝐴)) ErALTV 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 394   = wceq 1533  cin 3940   E cep 5576  ccnv 5672  dom cdm 5673  cres 5675   / cqs 8717  ccoss 37701   EqvRel weqvrel 37718   ErALTV werALTV 37727   Disj wdisjALTV 37735   Part wpart 37740
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 2166  ax-ext 2696  ax-sep 5295  ax-nul 5302  ax-pr 5424
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rmo 3364  df-rab 3420  df-v 3465  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4320  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5145  df-opab 5207  df-id 5571  df-eprel 5577  df-xp 5679  df-rel 5680  df-cnv 5681  df-co 5682  df-dm 5683  df-rn 5684  df-res 5685  df-ima 5686  df-ec 8720  df-qs 8724  df-coss 37935  df-refrel 38036  df-cnvrefrel 38051  df-symrel 38068  df-trrel 38098  df-eqvrel 38109  df-dmqs 38163  df-erALTV 38188  df-funALTV 38206  df-disjALTV 38229  df-eldisj 38231  df-part 38290
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator