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Theorem elcoeleqvrels 36995
Description: Elementhood in the coelement equivalence relations class. (Contributed by Peter Mazsa, 24-Jul-2023.)
Assertion
Ref Expression
elcoeleqvrels (𝐴𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ ( E ↾ 𝐴) ∈ EqvRels ))

Proof of Theorem elcoeleqvrels
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 reseq2 5930 . . . 4 (𝑎 = 𝐴 → ( E ↾ 𝑎) = ( E ↾ 𝐴))
21cosseqd 36828 . . 3 (𝑎 = 𝐴 → ≀ ( E ↾ 𝑎) = ≀ ( E ↾ 𝐴))
32eleq1d 2822 . 2 (𝑎 = 𝐴 → ( ≀ ( E ↾ 𝑎) ∈ EqvRels ↔ ≀ ( E ↾ 𝐴) ∈ EqvRels ))
4 df-coeleqvrels 36986 . 2 CoElEqvRels = {𝑎 ∣ ≀ ( E ↾ 𝑎) ∈ EqvRels }
53, 4elab2g 3630 1 (𝐴𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ ( E ↾ 𝐴) ∈ EqvRels ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205   = wceq 1541  wcel 2106   E cep 5534  ccnv 5630  cres 5633  ccoss 36572   EqvRels ceqvrels 36588   CoElEqvRels ccoeleqvrels 36590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-rab 3406  df-in 3915  df-br 5104  df-opab 5166  df-xp 5637  df-res 5643  df-coss 36811  df-coeleqvrels 36986
This theorem is referenced by:  elcoeleqvrelsrel  36996
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