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Theorem dfcoeleqvrels 36009
Description: Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 36007, eqvrelcoss3 36006 and eqvrelcoss4 36008 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)
Assertion
Ref Expression
dfcoeleqvrels CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }

Proof of Theorem dfcoeleqvrels
StepHypRef Expression
1 df-coeleqvrels 35974 . 2 CoElEqvRels = {𝑎 ∣ ≀ ( E ↾ 𝑎) ∈ EqvRels }
2 df-coels 35813 . . . 4 𝑎 = ≀ ( E ↾ 𝑎)
32eleq1i 2883 . . 3 ( ∼ 𝑎 ∈ EqvRels ↔ ≀ ( E ↾ 𝑎) ∈ EqvRels )
43abbii 2866 . 2 {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } = {𝑎 ∣ ≀ ( E ↾ 𝑎) ∈ EqvRels }
51, 4eqtr4i 2827 1 CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }
Colors of variables: wff setvar class
Syntax hints:   = wceq 1538  wcel 2112  {cab 2779   E cep 5432  ccnv 5522  cres 5525  ccoss 35606  ccoels 35607   EqvRels ceqvrels 35622   CoElEqvRels ccoeleqvrels 35624
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-ext 2773
This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2780  df-cleq 2794  df-clel 2873  df-coels 35813  df-coeleqvrels 35974
This theorem is referenced by: (None)
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