Theorem dfcoeleqvrels 38145

Description: Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 38143, eqvrelcoss3 38142 and eqvrelcoss4 38144 when needed. (Contributed by Peter Mazsa, 28-Nov-2022.)

Assertion

Ref	Expression
dfcoeleqvrels	⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }

Proof of Theorem dfcoeleqvrels

Step	Hyp	Ref	Expression
1		df-coeleqvrels 38110	. 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
2		df-coels 37936	. . . 4 ⊢ ∼ 𝑎 = ≀ (◡ E ↾ 𝑎)
3	2	eleq1i 2816	. . 3 ⊢ ( ∼ 𝑎 ∈ EqvRels ↔ ≀ (◡ E ↾ 𝑎) ∈ EqvRels )
4	3	abbii 2795	. 2 ⊢ {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
5	1, 4	eqtr4i 2756	1 ⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }

Syntax hints:   = wceq 1533   ∈ wcel 2098  {cab 2702   E cep 5576  ^◡ccnv 5672   ↾ cres 5675   ≀ ccoss 37701   ∼ ccoels 37702   EqvRels ceqvrels 37717   CoElEqvRels ccoeleqvrels 37719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696

This theorem depends on definitions:  df-bi 206  df-an 395  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-coels 37936  df-coeleqvrels 38110

This theorem is referenced by: (None)