Theorem eleldisjs 39210

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 5933	. . 3 ⊢ (𝑎 = 𝐴 → (◡ E ↾ 𝑎) = (◡ E ↾ 𝐴))
2	1	eleq1d 2826	. 2 ⊢ (𝑎 = 𝐴 → ((◡ E ↾ 𝑎) ∈ Disjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))
3		df-eldisjs 39173	. 2 ⊢ ElDisjs = {𝑎 ∣ (◡ E ↾ 𝑎) ∈ Disjs }
4	2, 3	elab2g 3620	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ ElDisjs ↔ (◡ E ↾ 𝐴) ∈ Disjs ))

Syntax hints:   → wi 4   ↔ wb 208   = wceq 1548   ∈ wcel 2121   E cep 5520  ^◡ccnv 5620   ↾ cres 5623   Disjs cdisjs 38600   ElDisjs celdisjs 38602

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-tru 1551  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-rab 3394  df-in 3892  df-opab 5138  df-xp 5627  df-res 5633  df-eldisjs 39173

This theorem is referenced by:  eleldisjseldisj  39211