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Theorem eleldisjs 38707
Description: Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.)
Assertion
Ref Expression
eleldisjs (𝐴𝑉 → (𝐴 ∈ ElDisjs ↔ ( E ↾ 𝐴) ∈ Disjs ))

Proof of Theorem eleldisjs
Dummy variable 𝑎 is distinct from all other variables.
StepHypRef Expression
1 reseq2 5990 . . 3 (𝑎 = 𝐴 → ( E ↾ 𝑎) = ( E ↾ 𝐴))
21eleq1d 2825 . 2 (𝑎 = 𝐴 → (( E ↾ 𝑎) ∈ Disjs ↔ ( E ↾ 𝐴) ∈ Disjs ))
3 df-eldisjs 38685 . 2 ElDisjs = {𝑎 ∣ ( E ↾ 𝑎) ∈ Disjs }
42, 3elab2g 3679 1 (𝐴𝑉 → (𝐴 ∈ ElDisjs ↔ ( E ↾ 𝐴) ∈ Disjs ))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1540  wcel 2108   E cep 5581  ccnv 5682  cres 5685   Disjs cdisjs 38193   ElDisjs celdisjs 38195
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 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2728  df-clel 2815  df-rab 3436  df-in 3957  df-opab 5204  df-xp 5689  df-res 5695  df-eldisjs 38685
This theorem is referenced by:  eleldisjseldisj  38708
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