Theorem dfcoeleqvrels 39456

Description: Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 39454, eqvrelcoss3 39453 and eqvrelcoss4 39455 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 39421	. 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
2		df-coels 39253	. . . 4 ⊢ ∼ 𝑎 = ≀ (◡ E ↾ 𝑎)
3	2	eleq1i 2851	. . 3 ⊢ ( ∼ 𝑎 ∈ EqvRels ↔ ≀ (◡ E ↾ 𝑎) ∈ EqvRels )
4	3	abbii 2827	. 2 ⊢ {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
5	1, 4	eqtr4i 2786	1 ⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }

This proof depends on syntax axioms:   = wceq 1570   ∈ wcel 2145  {cab 2738   E cep 5554  ^◡ccnv 5654   ↾ cres 5657   ≀ ccoss 38934   ∼ ccoels 38935   EqvRels ceqvrels 38950   CoElEqvRels ccoeleqvrels 38952

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-coels 39253  df-coeleqvrels 39421

This theorem is used by: (None)