Theorem bj-ismoore 37775

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 37772 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 3475	. 2 ⊢ (𝐴 ∈ Moore → 𝐴 ∈ V)
2		bj-mooreset 37772	. 2 ⊢ (∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴 → 𝐴 ∈ V)
3		pweq 4575	. . . 4 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
4		unieq 4882	. . . . . 6 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
5	4	ineq1d 4171	. . . . 5 ⊢ (𝑦 = 𝐴 → (∪ 𝑦 ∩ ∩ 𝑥) = (∪ 𝐴 ∩ ∩ 𝑥))
6		id 23	. . . . 5 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
7	5, 6	eleq12d 2856	. . . 4 ⊢ (𝑦 = 𝐴 → ((∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦 ↔ (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
8	3, 7	raleqbidv 3337	. . 3 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝒫 𝑦(∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
9		df-bj-moore 37774	. . 3 ⊢ Moore = {𝑦 ∣ ∀𝑥 ∈ 𝒫 𝑦(∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦}
10	8, 9	elab2g 3638	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
11	1, 2, 10	pm5.21nii 381	1 ⊢ (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1569   ∈ wcel 2142  ∀wral 3078  Vcvv 3454   ∩ cin 3903  𝒫 cpw 4561  ∪ cuni 4871  ∩ cint 4911  Moorecmoore 37773

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734  ax-sep 5256  ax-nul 5268  ax-pow 5335

This proof depends on definitions:  df-bi 210  df-an 401  df-3an 1104  df-tru 1572  df-fal 1582  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3416  df-v 3456  df-dif 3907  df-in 3911  df-ss 3921  df-nul 4286  df-pw 4563  df-uni 4872  df-int 4912  df-bj-moore 37774

This theorem is used by:  bj-ismoored0  37776  bj-ismoored  37777  bj-ismooredr  37779