Theorem dfcoeleqvrels 36841

Description: Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 36839, eqvrelcoss3 36838 and eqvrelcoss4 36840 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 36806	. 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
2		df-coels 36632	. . . 4 ⊢ ∼ 𝑎 = ≀ (◡ E ↾ 𝑎)
3	2	eleq1i 2827	. . 3 ⊢ ( ∼ 𝑎 ∈ EqvRels ↔ ≀ (◡ E ↾ 𝑎) ∈ EqvRels )
4	3	abbii 2806	. 2 ⊢ {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
5	1, 4	eqtr4i 2767	1 ⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }

Syntax hints:   = wceq 1539   ∈ wcel 2104  {cab 2713   E cep 5505  ^◡ccnv 5599   ↾ cres 5602   ≀ ccoss 36387   ∼ ccoels 36388   EqvRels ceqvrels 36403   CoElEqvRels ccoeleqvrels 36405

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 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-coels 36632  df-coeleqvrels 36806

This theorem is referenced by: (None)