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Theorem petid 38289
Description: A class is a partition by the identity class if and only if the cosets by the identity class are in equivalence relation on it. (Contributed by Peter Mazsa, 31-Dec-2021.)
Assertion
Ref Expression
petid ( I Part 𝐴 ↔ ≀ I ErALTV 𝐴)

Proof of Theorem petid
StepHypRef Expression
1 petid2 38288 . 2 (( Disj I ∧ (dom I / I ) = 𝐴) ↔ ( EqvRel ≀ I ∧ (dom ≀ I / ≀ I ) = 𝐴))
2 dfpart2 38241 . 2 ( I Part 𝐴 ↔ ( Disj I ∧ (dom I / I ) = 𝐴))
3 dferALTV2 38140 . 2 ( ≀ I ErALTV 𝐴 ↔ ( EqvRel ≀ I ∧ (dom ≀ I / ≀ I ) = 𝐴))
41, 2, 33bitr4i 303 1 ( I Part 𝐴 ↔ ≀ I ErALTV 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395   = wceq 1534   I cid 5575  dom cdm 5678   / cqs 8723  ccoss 37648   EqvRel weqvrel 37665   ErALTV werALTV 37674   Disj wdisjALTV 37682   Part wpart 37687
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 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
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 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rmo 3373  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8726  df-qs 8730  df-coss 37883  df-refrel 37984  df-cnvrefrel 37999  df-symrel 38016  df-trrel 38046  df-eqvrel 38057  df-dmqs 38111  df-erALTV 38136  df-disjALTV 38177  df-part 38238
This theorem is referenced by: (None)
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