Theorem eleldisjs 39427

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 5977	. . 3 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴))
2	1	eleq1d 2855	. 2 ⊢ (𝑎 = 𝐴 → ((◡ E ↾ 𝑎) ∈ Disjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))
3		df-eldisjs 39390	. 2 ⊢ ElDisjs = {𝑎 ∣ (◡ E ↾ 𝑎) ∈ Disjs }
4	2, 3	elab2g 3647	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1568   ∈ wcel 2150   E cep 5564  ^◡ccnv 5664   ↾ cres 5667   Disjs cdisjs 38817   ElDisjs celdisjs 38819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-ex 1808  df-sb 2099  df-clab 2749  df-cleq 2762  df-clel 2845  df-rab 3424  df-in 3920  df-opab 5179  df-xp 5671  df-res 5677  df-eldisjs 39390

This theorem is referenced by:  eleldisjseldisj  39428