Theorem bj-ismoore 37100

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 37097 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 3471	. 2 ⊢ (𝐴 ∈ Moore → 𝐴 ∈ V)
2		bj-mooreset 37097	. 2 ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V)
3		pweq 4580	. . . 4 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
4		unieq 4885	. . . . . 6 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
5	4	ineq1d 4185	. . . . 5 ⊢ (𝑦 = 𝐴 → (∪ 𝑦 ∩ ∩ 𝑥) = (∪ 𝐴 ∩ ∩ 𝑥))
6		id 22	. . . . 5 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
7	5, 6	eleq12d 2823	. . . 4 ⊢ (𝑦 = 𝐴 → ((∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦 ↔ (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
8	3, 7	raleqbidv 3321	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝒫 𝑦(∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
9		df-bj-moore 37099	. . 3 ⊢ Moore = {𝑦 ∣ ∀𝑥 ∈ 𝒫 𝑦(∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦}
10	8, 9	elab2g 3650	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
11	1, 2, 10	pm5.21nii 378	1 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)

Syntax hints:   ↔ wb 206   = wceq 1540   ∈ wcel 2109  ∀wral 3045  Vcvv 3450   ∩ cin 3916  𝒫 cpw 4566  ∪ cuni 4874  ∩ cint 4913  Moorecmoore 37098

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323

This theorem depends on definitions:  df-bi 207  df-an 396  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-in 3924  df-ss 3934  df-nul 4300  df-pw 4568  df-uni 4875  df-int 4914  df-bj-moore 37099

This theorem is referenced by:  bj-ismoored0  37101  bj-ismoored  37102  bj-ismooredr  37104