Theorem elcoeleqvrels 38697

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 5928	. . . 4 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴))
2	1	cosseqd 38536	. . 3 ⊢ (𝑎 = 𝐴 → ≀ (◡ E ↾ 𝑎) = ≀ (◡ E ↾ 𝐴))
3	2	eleq1d 2816	. 2 ⊢ (𝑎 = 𝐴 → ( ≀ (◡ E ↾ 𝑎) ∈ EqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))
4		df-coeleqvrels 38688	. 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
5	3, 4	elab2g 3631	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111   E cep 5518  ^◡ccnv 5618   ↾ cres 5621   ≀ ccoss 38228   EqvRels ceqvrels 38244   CoElEqvRels ccoeleqvrels 38246

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-in 3904  df-br 5094  df-opab 5156  df-xp 5625  df-res 5631  df-coss 38519  df-coeleqvrels 38688

This theorem is referenced by:  elcoeleqvrelsrel  38698