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Theorem detinidres 38179
Description: The cosets by the intersection with the restricted identity relation are in equivalence relation if and only if the intersection with the restricted identity relation is disjoint. (Contributed by Peter Mazsa, 31-Dec-2021.)
Assertion
Ref Expression
detinidres ( Disj (𝑅 ∩ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴)))

Proof of Theorem detinidres
StepHypRef Expression
1 disjALTVinidres 38140 . 2 Disj (𝑅 ∩ ( I ↾ 𝐴))
21detlem 38166 1 ( Disj (𝑅 ∩ ( I ↾ 𝐴)) ↔ EqvRel ≀ (𝑅 ∩ ( I ↾ 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wb 205  cin 3942   I cid 5566  cres 5671  ccoss 37556   EqvRel weqvrel 37573   Disj wdisjALTV 37590
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  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 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-coss 37794  df-refrel 37895  df-cnvrefrel 37910  df-symrel 37927  df-trrel 37957  df-eqvrel 37968  df-funALTV 38065  df-disjALTV 38088
This theorem is referenced by: (None)
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