Theorem elcoeleqvrels 37465

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 5977	. . . 4 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴))
2	1	cosseqd 37298	. . 3 ⊢ (𝑎 = 𝐴 → ≀ (◡ E ↾ 𝑎) = ≀ (◡ E ↾ 𝐴))
3	2	eleq1d 2819	. 2 ⊢ (𝑎 = 𝐴 → ( ≀ (◡ E ↾ 𝑎) ∈ EqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))
4		df-coeleqvrels 37456	. 2 ⊢ CoElEqvRels = {𝑎 ∣ ≀ (◡ E ↾ 𝑎) ∈ EqvRels }
5	3, 4	elab2g 3671	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ (◡ E ↾ 𝐴) ∈ EqvRels ))

Syntax hints:   → wi 4   ↔ wb 205   = wceq 1542   ∈ wcel 2107   E cep 5580  ^◡ccnv 5676   ↾ cres 5679   ≀ ccoss 37043   EqvRels ceqvrels 37059   CoElEqvRels ccoeleqvrels 37061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-in 3956  df-br 5150  df-opab 5212  df-xp 5683  df-res 5689  df-coss 37281  df-coeleqvrels 37456

This theorem is referenced by:  elcoeleqvrelsrel  37466