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Theorem petincnvepres 38850
Description: The shortest form of a partition-equivalence theorem with intersection and general 𝑅. Cf. br1cossincnvepres 38451. Cf. pet 38852. (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 38849 . 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 38770 . 2 ((𝑅 ∩ ( E ↾ 𝐴)) Part 𝐴 ↔ ( Disj (𝑅 ∩ ( E ↾ 𝐴)) ∧ (dom (𝑅 ∩ ( E ↾ 𝐴)) / (𝑅 ∩ ( E ↾ 𝐴))) = 𝐴))
3 dferALTV2 38669 . 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 1540  cin 3950   E cep 5583  ccnv 5684  dom cdm 5685  cres 5687   / cqs 8744  ccoss 38182   EqvRel weqvrel 38199   ErALTV werALTV 38208   Disj wdisjALTV 38216   Part wpart 38221
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-qs 8751  df-coss 38412  df-refrel 38513  df-cnvrefrel 38528  df-symrel 38545  df-trrel 38575  df-eqvrel 38586  df-dmqs 38640  df-erALTV 38665  df-funALTV 38683  df-disjALTV 38706  df-eldisj 38708  df-part 38767
This theorem is referenced by: (None)
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