Definition df-acs 16249

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 8540 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 16245	. 2 class ACS
2		vx	. . 3 setvar 𝑥
3		cvv 3200	. . 3 class V
4	2	cv 1482	. . . . . . . 8 class 𝑥
5	4	cpw 4158	. . . . . . 7 class 𝒫 𝑥
6		vf	. . . . . . . 8 setvar 𝑓
7	6	cv 1482	. . . . . . 7 class 𝑓
8	5, 5, 7	wf 5884	. . . . . 6 wff 𝑓:𝒫 𝑥⟶𝒫 𝑥
9		vs	. . . . . . . . 9 setvar 𝑠
10		vc	. . . . . . . . 9 setvar 𝑐
11	9, 10	wel 1991	. . . . . . . 8 wff 𝑠 ∈ 𝑐
12	9	cv 1482	. . . . . . . . . . . . 13 class 𝑠
13	12	cpw 4158	. . . . . . . . . . . 12 class 𝒫 𝑠
14		cfn 7955	. . . . . . . . . . . 12 class Fin
15	13, 14	cin 3573	. . . . . . . . . . 11 class (𝒫 𝑠 ∩ Fin)
16	7, 15	cima 5117	. . . . . . . . . 10 class (𝑓 “ (𝒫 𝑠 ∩ Fin))
17	16	cuni 4436	. . . . . . . . 9 class ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin))
18	17, 12	wss 3574	. . . . . . . 8 wff ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠
19	11, 18	wb 196	. . . . . . 7 wff (𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
20	19, 9, 5	wral 2912	. . . . . 6 wff ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
21	8, 20	wa 384	. . . . 5 wff (𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
22	21, 6	wex 1704	. . . 4 wff ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
23		cmre 16242	. . . . 5 class Moore
24	4, 23	cfv 5888	. . . 4 class (Moore‘𝑥)
25	22, 10, 24	crab 2916	. . 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 4729	. 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 1483	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  16312