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Theorem dfcoeleqvrels 38030
Description: Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 38028, eqvrelcoss3 38027 and eqvrelcoss4 38029 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 37995 . 2 CoElEqvRels = {𝑎 ∣ ≀ ( E ↾ 𝑎) ∈ EqvRels }
2 df-coels 37821 . . . 4 𝑎 = ≀ ( E ↾ 𝑎)
32eleq1i 2819 . . 3 ( ∼ 𝑎 ∈ EqvRels ↔ ≀ ( E ↾ 𝑎) ∈ EqvRels )
43abbii 2797 . 2 {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } = {𝑎 ∣ ≀ ( E ↾ 𝑎) ∈ EqvRels }
51, 4eqtr4i 2758 1 CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }
Colors of variables: wff setvar class
Syntax hints:   = wceq 1534  wcel 2099  {cab 2704   E cep 5575  ccnv 5671  cres 5674  ccoss 37583  ccoels 37584   EqvRels ceqvrels 37599   CoElEqvRels ccoeleqvrels 37601
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 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-coels 37821  df-coeleqvrels 37995
This theorem is referenced by: (None)
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