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Theorem brers 38139
Description: Binary equivalence relation with natural domain, see the comment of df-ers 38135. (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 38135 . 2 Ers = ( DomainQss ↾ EqvRels )
21eqres 37812 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 2099   class class class wbr 5148   EqvRels ceqvrels 37664   DomainQss cdmqss 37671   Ers cers 37673
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-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-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  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-xp 5684  df-res 5690  df-ers 38135
This theorem is referenced by:  brerser  38149
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