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Theorem brparts 37164
Description: Binary partitions relation. (Contributed by Peter Mazsa, 23-Jul-2021.)
Assertion
Ref Expression
brparts (𝐴𝑉 → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴)))

Proof of Theorem brparts
StepHypRef Expression
1 df-parts 37158 . 2 Parts = ( DomainQss ↾ Disjs )
21eqres 36732 1 (𝐴𝑉 → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106   class class class wbr 5103   DomainQss cdmqss 36588   Disjs cdisjs 36598   Parts cparts 36603
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-res 5643  df-parts 37158
This theorem is referenced by:  brparts2  37165  brpartspart  37166
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