Theorem bj-ismoore 33391

Description: Characterization of Moore collections among sets. (Contributed by BJ, 9-Dec-2021.)

Assertion

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

Distinct variable group:   𝑥,𝐴

Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem bj-ismoore

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		pweq 4300	. . 3 ⊢ (𝑦 = 𝐴 → 𝒫 𝑦 = 𝒫 𝐴)
2		unieq 4582	. . . . 5 ⊢ (𝑦 = 𝐴 → ∪ 𝑦 = ∪ 𝐴)
3	2	ineq1d 3964	. . . 4 ⊢ (𝑦 = 𝐴 → (∪ 𝑦 ∩ ∩ 𝑥) = (∪ 𝐴 ∩ ∩ 𝑥))
4		id 22	. . . 4 ⊢ (𝑦 = 𝐴 → 𝑦 = 𝐴)
5	3, 4	eleq12d 2844	. . 3 ⊢ (𝑦 = 𝐴 → ((∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦 ↔ (∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
6	1, 5	raleqbidv 3301	. 2 ⊢ (𝑦 = 𝐴 → (∀𝑥 ∈ 𝒫 𝑦(∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦 ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))
7		df-bj-moore 33390	. 2 ⊢ Moore = {𝑦 ∣ ∀𝑥 ∈ 𝒫 𝑦(∪ 𝑦 ∩ ∩ 𝑥) ∈ 𝑦}
8	6, 7	elab2g 3504	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ Moore ↔ ∀𝑥 ∈ 𝒫 𝐴(∪ 𝐴 ∩ ∩ 𝑥) ∈ 𝐴))

Syntax hints:   → wi 4   ↔ wb 196   = wceq 1631   ∈ wcel 2145  ∀wral 3061   ∩ cin 3722  𝒫 cpw 4297  ∪ cuni 4574  ∩ cint 4611  Moorecmoore 33389

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-v 3353  df-in 3730  df-ss 3737  df-pw 4299  df-uni 4575  df-bj-moore 33390

This theorem is referenced by:  bj-ismoorec  33392  bj-ismoored0  33393  bj-ismooredr  33396  bj-ismooredr2  33397