Theorem bj-ismoore 34917

Description: Characterization of Moore collections. Note that there is no sethood hypothesis on 𝐴: it is implied by either side (this is obvious for the LHS, and is the content of bj-mooreset 34914 for the RHS). (Contributed by BJ, 9-Dec-2021.)

Assertion

Ref	Expression
bj-ismoore	⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem bj-ismoore

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elex 3416	. 2 ⊢ (𝐴 ∈ Moore → 𝐴 ∈ V)
2		bj-mooreset 34914	. 2 ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V)
3		pweq 4504	. . . 4 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
4		unieq 4807	. . . . . 6 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
5	4	ineq1d 4102	. . . . 5 ⊢ (𝑦 = 𝐴 → (∪ 𝑦 ∩ ∩ 𝑥) = (∪ 𝐴 ∩ ∩ 𝑥))
6		id 22	. . . . 5 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
7	5, 6	eleq12d 2827	. . . 4 ⊢ (𝑦 = 𝐴 → ((∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦 ↔ (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
8	3, 7	raleqbidv 3304	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝒫 𝑦(∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
9		df-bj-moore 34916	. . 3 ⊢ Moore = {𝑦 ∣ ∀𝑥 ∈ 𝒫 𝑦(∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦}
10	8, 9	elab2g 3575	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
11	1, 2, 10	pm5.21nii 383	1 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)

Syntax hints:   ↔ wb 209   = wceq 1542   ∈ wcel 2114  ∀wral 3053  Vcvv 3398   ∩ cin 3842  𝒫 cpw 4488  ∪ cuni 4796  ∩ cint 4836  Moorecmoore 34915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-ext 2710  ax-sep 5167  ax-nul 5174  ax-pow 5232

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1545  df-fal 1555  df-ex 1787  df-sb 2075  df-clab 2717  df-cleq 2730  df-clel 2811  df-ral 3058  df-rab 3062  df-v 3400  df-dif 3846  df-in 3850  df-ss 3860  df-nul 4212  df-pw 4490  df-uni 4797  df-int 4837  df-bj-moore 34916

This theorem is referenced by:  bj-ismoored0  34918  bj-ismoored  34919  bj-ismooredr  34921