Theorem dfcoeleqvrels 38093

Description: Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 38091, eqvrelcoss3 38090 and eqvrelcoss4 38092 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 38058	. 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
2		df-coels 37884	. . . 4 ⊢ ∼ 𝑎 = ≀ (◡ E ↾ 𝑎)
3	2	eleq1i 2820	. . 3 ⊢ ( ∼ 𝑎 ∈ EqvRels ↔ ≀ (◡ E ↾ 𝑎) ∈ EqvRels )
4	3	abbii 2798	. 2 ⊢ {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
5	1, 4	eqtr4i 2759	1 ⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }

Syntax hints:   = wceq 1534   ∈ wcel 2099  {cab 2705   E cep 5581  ^◡ccnv 5677   ↾ cres 5680   ≀ ccoss 37648   ∼ ccoels 37649   EqvRels ceqvrels 37664   CoElEqvRels ccoeleqvrels 37666

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 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-coels 37884  df-coeleqvrels 38058

This theorem is referenced by: (None)