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Theorem brers 37525
Description: Binary equivalence relation with natural domain, see the comment of df-ers 37521. (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 37521 . 2 Ers = ( DomainQss ↾ EqvRels )
21eqres 37197 1 (𝐴𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wcel 2106   class class class wbr 5147   EqvRels ceqvrels 37047   DomainQss cdmqss 37054   Ers cers 37056
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 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
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 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-res 5687  df-ers 37521
This theorem is referenced by:  brerser  37535
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