Theorem elcoeleqvrels 35863

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 5841	. . . 4 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴))
2	1	cosseqd 35706	. . 3 ⊢ (𝑎 = 𝐴 → ≀ (◡ E ↾ 𝑎) = ≀ (◡ E ↾ 𝐴))
3	2	eleq1d 2896	. 2 ⊢ (𝑎 = 𝐴 → ( ≀ (◡ E ↾ 𝑎) ∈ EqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))
4		df-coeleqvrels 35854	. 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
5	3, 4	elab2g 3664	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1536   ∈ wcel 2113   E cep 5457  ^◡ccnv 5547   ↾ cres 5550   ≀ ccoss 35486   EqvRels ceqvrels 35502   CoElEqvRels ccoeleqvrels 35504

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1969  ax-7 2014  ax-8 2115  ax-9 2123  ax-10 2144  ax-11 2160  ax-12 2176  ax-ext 2792

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1539  df-ex 1780  df-nf 1784  df-sb 2069  df-clab 2799  df-cleq 2813  df-clel 2892  df-nfc 2962  df-rab 3146  df-in 3936  df-br 5060  df-opab 5122  df-xp 5554  df-res 5560  df-coss 35692  df-coeleqvrels 35854

This theorem is referenced by:  elcoeleqvrelsrel  35864