Theorem brparts 39491

Description: Binary partitions relation. (Contributed by Peter Mazsa, 23-Jul-2021.)

Assertion

Ref	Expression
brparts	⊢ (𝐴 ∈ 𝑉 → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴)))

Proof of Theorem brparts

Step	Hyp	Ref	Expression
1		df-parts 39485	. 2 ⊢ Parts = ( DomainQss ↾ Disjs )
2	1	eqres 38957	1 ⊢ (𝐴 ∈ 𝑉 → (𝑅 Parts 𝐴 ↔ (𝑅 ∈ Disjs ∧ 𝑅 DomainQss 𝐴)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   ∈ wcel 2141   class class class wbr 5108   DomainQss cdmqss 38823   Disjs cdisjs 38835   Parts cparts 38840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-br 5109  df-opab 5173  df-xp 5667  df-res 5673  df-parts 39485

This theorem is referenced by:  brparts2  39492  brpartspart  39493