Theorem brers 39090

Description: Binary equivalence relation with natural domain, see the comment of df-ers 39086. (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 39086	. 2 ⊢ Ers = ( DomainQss ↾ EqvRels )
2	1	eqres 38678	1 ⊢ (𝐴 ∈ 𝑉 → (𝑅 Ers 𝐴 ↔ (𝑅 ∈ EqvRels ∧ 𝑅 DomainQss 𝐴)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2114   class class class wbr 5086   EqvRels ceqvrels 38537   DomainQss cdmqss 38544   Ers cers 38546

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5371

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5631  df-res 5637  df-ers 39086

This theorem is referenced by:  brerser  39100  dfpeters2  39312