Definition df-acs 17557

Description: An important subclass of Moore systems are those which can be interpreted as closure under some collection of operators of finite arity (the collection itself is not required to be finite). These are termed algebraic closure systems; similar to definition (A) of an algebraic closure system in [Schechter] p. 84, but to avoid the complexity of an arbitrary mixed collection of functions of various arities (especially if the axiom of infinity omex 9603 is to be avoided), we consider a single function defined on finite sets instead. (Contributed by Stefan O'Rear, 2-Apr-2015.)

Assertion

Ref	Expression
df-acs	⊢ ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})

Distinct variable group:   𝑓,𝑐,𝑠,𝑥

Detailed syntax breakdown of Definition df-acs

Step	Hyp	Ref	Expression
1		cacs 17553	. 2 class ACS
2		vx	. . 3 setvar 𝑥
3		cvv 3450	. . 3 class V
4	2	cv 1539	. . . . . . . 8 class 𝑥
5	4	cpw 4566	. . . . . . 7 class 𝒫 𝑥
6		vf	. . . . . . . 8 setvar 𝑓
7	6	cv 1539	. . . . . . 7 class 𝑓
8	5, 5, 7	wf 6510	. . . . . 6 wff 𝑓:𝒫 𝑥⟶𝒫 𝑥
9		vs	. . . . . . . . 9 setvar 𝑠
10		vc	. . . . . . . . 9 setvar 𝑐
11	9, 10	wel 2110	. . . . . . . 8 wff 𝑠 ∈ 𝑐
12	9	cv 1539	. . . . . . . . . . . . 13 class 𝑠
13	12	cpw 4566	. . . . . . . . . . . 12 class 𝒫 𝑠
14		cfn 8921	. . . . . . . . . . . 12 class Fin
15	13, 14	cin 3916	. . . . . . . . . . 11 class (𝒫 𝑠 ∩ Fin)
16	7, 15	cima 5644	. . . . . . . . . 10 class (𝑓 “ (𝒫 𝑠 ∩ Fin))
17	16	cuni 4874	. . . . . . . . 9 class ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin))
18	17, 12	wss 3917	. . . . . . . 8 wff ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠
19	11, 18	wb 206	. . . . . . 7 wff (𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
20	19, 9, 5	wral 3045	. . . . . 6 wff ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
21	8, 20	wa 395	. . . . 5 wff (𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
22	21, 6	wex 1779	. . . 4 wff ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
23		cmre 17550	. . . . 5 class Moore
24	4, 23	cfv 6514	. . . 4 class (Moore‘𝑥)
25	22, 10, 24	crab 3408	. . 3 class {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))}
26	2, 3, 25	cmpt 5191	. 2 class (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
27	1, 26	wceq 1540	1 wff ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})

This definition is referenced by:  isacs  17619