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Theorem elcoeleqvrels 38551
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 6004 . . . 4 (𝑎 = 𝐴 → ( E ↾ 𝑎) = ( E ↾ 𝐴))
21cosseqd 38384 . . 3 (𝑎 = 𝐴 → ≀ ( E ↾ 𝑎) = ≀ ( E ↾ 𝐴))
32eleq1d 2829 . 2 (𝑎 = 𝐴 → ( ≀ ( E ↾ 𝑎) ∈ EqvRels ↔ ≀ ( E ↾ 𝐴) ∈ EqvRels ))
4 df-coeleqvrels 38542 . 2 CoElEqvRels = {𝑎 ∣ ≀ ( E ↾ 𝑎) ∈ EqvRels }
53, 4elab2g 3696 1 (𝐴𝑉 → (𝐴 ∈ CoElEqvRels ↔ ≀ ( E ↾ 𝐴) ∈ EqvRels ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2108   E cep 5598  ccnv 5699  cres 5702  ccoss 38135   EqvRels ceqvrels 38151   CoElEqvRels ccoeleqvrels 38153
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-rab 3444  df-in 3983  df-br 5167  df-opab 5229  df-xp 5706  df-res 5712  df-coss 38367  df-coeleqvrels 38542
This theorem is referenced by:  elcoeleqvrelsrel  38552
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