Theorem eleldisjs 38671

Description: Elementhood in the disjoint elements class. (Contributed by Peter Mazsa, 23-Jul-2023.)

Assertion

Ref	Expression
eleldisjs	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))

Proof of Theorem eleldisjs

Dummy variable 𝑎 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		reseq2 5989	. . 3 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴))
2	1	eleq1d 2822	. 2 ⊢ (𝑎 = 𝐴 → ((◡ E ↾ 𝑎) ∈ Disjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))
3		df-eldisjs 38649	. 2 ⊢ ElDisjs = {𝑎 ∣ (◡ E ↾ 𝑎) ∈ Disjs }
4	2, 3	elab2g 3683	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1535   ∈ wcel 2104   E cep 5581  ^◡ccnv 5682   ↾ cres 5685   Disjs cdisjs 38155   ElDisjs celdisjs 38157

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1963  ax-7 2003  ax-8 2106  ax-9 2114  ax-ext 2704

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1538  df-ex 1775  df-sb 2061  df-clab 2711  df-cleq 2725  df-clel 2812  df-rab 3433  df-in 3970  df-opab 5212  df-xp 5689  df-res 5695  df-eldisjs 38649

This theorem is referenced by:  eleldisjseldisj  38672