Theorem eleldisjs 38836

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 5922	. . 3 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴))
2	1	eleq1d 2816	. 2 ⊢ (𝑎 = 𝐴 → ((◡ E ↾ 𝑎) ∈ Disjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))
3		df-eldisjs 38814	. 2 ⊢ ElDisjs = {𝑎 ∣ (◡ E ↾ 𝑎) ∈ Disjs }
4	2, 3	elab2g 3631	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ∈ wcel 2111   E cep 5513  ^◡ccnv 5613   ↾ cres 5616   Disjs cdisjs 38265   ElDisjs celdisjs 38267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-rab 3396  df-in 3904  df-opab 5152  df-xp 5620  df-res 5626  df-eldisjs 38814

This theorem is referenced by:  eleldisjseldisj  38837