Theorem dfcoeleqvrels 39777

Description: Alternate definition of the coelement equivalence relations class. Other alternate definitions should be based on eqvrelcoss2 39775, eqvrelcoss3 39774 and eqvrelcoss4 39776 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 39742	. 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
2		df-coels 39574	. . . 4 ⊢ ∼ 𝑎 = ≀ (◡ E ↾ 𝑎)
3	2	eleq1i 2854	. . 3 ⊢ ( ∼ 𝑎 ∈ EqvRels ↔ ≀ (◡ E ↾ 𝑎) ∈ EqvRels )
4	3	abbii 2830	. 2 ⊢ {𝑎 ∣ ∼ 𝑎 ∈ EqvRels } = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
5	1, 4	eqtr4i 2789	1 ⊢ CoElEqvRels = {𝑎 ∣ ∼ 𝑎 ∈ EqvRels }

This proof depends on syntax axioms:   = wceq 1569   ∈ wcel 2146  {cab 2741   E cep 5553  ^◡ccnv 5653   ↾ cres 5656   ≀ ccoss 39255   ∼ ccoels 39256   EqvRels ceqvrels 39271   CoElEqvRels ccoeleqvrels 39273

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-coels 39574  df-coeleqvrels 39742

This theorem is used by: (None)