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Theorem dfcoeleqvrels 38622
Description: Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 38620, eqvrelcoss3 38619 and eqvrelcoss4 38621 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 38587 . 2 CoElEqvRels = {𝑎 ∣ ≀ ( E ↾ 𝑎) ∈ EqvRels }
2 df-coels 38413 . . . 4 𝑎 = ≀ ( E ↾ 𝑎)
32eleq1i 2832 . . 3 ( ∼ 𝑎 ∈ EqvRels ↔ ≀ ( E ↾ 𝑎) ∈ EqvRels )
43abbii 2809 . 2 {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } = {𝑎 ∣ ≀ ( E ↾ 𝑎) ∈ EqvRels }
51, 4eqtr4i 2768 1 CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }
Colors of variables: wff setvar class
Syntax hints:   = wceq 1540  wcel 2108  {cab 2714   E cep 5583  ccnv 5684  cres 5687  ccoss 38182  ccoels 38183   EqvRels ceqvrels 38198   CoElEqvRels ccoeleqvrels 38200
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-coels 38413  df-coeleqvrels 38587
This theorem is referenced by: (None)
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