Theorem elcoeleqvrels 38119

Description: Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023.)

Assertion

Ref	Expression
elcoeleqvrels	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))

Proof of Theorem elcoeleqvrels

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reseq2 5975	. . . 4 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴))
2	1	cosseqd 37952	. . 3 ⊢ (𝑎 = 𝐴 → ≀ (◡ E ↾ 𝑎) = ≀ (◡ E ↾ 𝐴))
3	2	eleq1d 2810	. 2 ⊢ (𝑎 = 𝐴 → ( ≀ (◡ E ↾ 𝑎) ∈ EqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))
4		df-coeleqvrels 38110	. 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
5	3, 4	elab2g 3663	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1533   ∈ wcel 2098   E cep 5576  ^◡ccnv 5672   ↾ cres 5675   ≀ ccoss 37701   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-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-rab 3420  df-in 3948  df-br 5145  df-opab 5207  df-xp 5679  df-res 5685  df-coss 37935  df-coeleqvrels 38110

This theorem is referenced by:  elcoeleqvrelsrel  38120