Theorem elcoeleqvrels 38577

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 5995	. . . 4 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴))
2	1	cosseqd 38410	. . 3 ⊢ (𝑎 = 𝐴 → ≀ (◡ E ↾ 𝑎) = ≀ (◡ E ↾ 𝐴))
3	2	eleq1d 2824	. 2 ⊢ (𝑎 = 𝐴 → ( ≀ (◡ E ↾ 𝑎) ∈ EqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))
4		df-coeleqvrels 38568	. 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
5	3, 4	elab2g 3683	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1537   ∈ wcel 2106   E cep 5588  ^◡ccnv 5688   ↾ cres 5691   ≀ ccoss 38162   EqvRels ceqvrels 38178   CoElEqvRels ccoeleqvrels 38180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-in 3970  df-br 5149  df-opab 5211  df-xp 5695  df-res 5701  df-coss 38393  df-coeleqvrels 38568

This theorem is referenced by:  elcoeleqvrelsrel  38578