Theorem brers 39950

Description: Binary equivalence relation with natural domain, see the comment of df-ers 39946. (Contributed by Peter Mazsa, 23-Jul-2021.)

Assertion

Ref	Expression
brers	⊢ (𝐴 ∈ 𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴)))

Proof of Theorem brers

Step	Hyp	Ref	Expression
1		df-ers 39946	. 2 ⊢ Ers = ( DomainQss ↾ EqvRels )
2	1	eqres 39538	1 ⊢ (𝐴 ∈ 𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∈ wcel 2147   class class class wbr 5108   EqvRels ceqvrels 39397   DomainQss cdmqss 39404   Ers cers 39406

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735  ax-sep 5255  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-br 5109  df-opab 5174  df-xp 5664  df-res 5670  df-ers 39946

This theorem is used by:  brerser  39960  dfpeters2  40172