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Theorem brers 38040
Description: Binary equivalence relation with natural domain, see the comment of df-ers 38036. (Contributed by Peter Mazsa, 23-Jul-2021.)
Assertion
Ref Expression
brers (𝐴𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴)))

Proof of Theorem brers
StepHypRef Expression
1 df-ers 38036 . 2 Ers = ( DomainQss ↾ EqvRels )
21eqres 37712 1 (𝐴𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wcel 2098   class class class wbr 5139   EqvRels ceqvrels 37562   DomainQss cdmqss 37569   Ers cers 37571
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-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
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-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-xp 5673  df-res 5679  df-ers 38036
This theorem is referenced by:  brerser  38050
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