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Theorem elcoeleqvrels 38577
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 5995 . . . 4 (𝑎 = 𝐴 → ( E ↾ 𝑎) = ( E ↾ 𝐴))
21cosseqd 38410 . . 3 (𝑎 = 𝐴 → ≀ ( E ↾ 𝑎) = ≀ ( E ↾ 𝐴))
32eleq1d 2824 . 2 (𝑎 = 𝐴 → ( ≀ ( E ↾ 𝑎) ∈ EqvRels ↔ ≀ ( E ↾ 𝐴) ∈ EqvRels ))
4 df-coeleqvrels 38568 . 2 CoElEqvRels = {𝑎 ∣ ≀ ( E ↾ 𝑎) ∈ EqvRels }
53, 4elab2g 3683 1 (𝐴𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ ( E ↾ 𝐴) ∈ EqvRels ))
Colors of variables: wff setvar class
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
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