Definition df-acs 17647

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 9610 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 17643	. 2 class ACS
2		vx	. . 3 setvar 𝑥
3		cvv 3454	. . 3 class V
4	2	cv 1568	. . . . . . . 8 class 𝑥
5	4	cpw 4561	. . . . . . 7 class 𝒫 𝑥
6		vf	. . . . . . . 8 setvar 𝑓
7	6	cv 1568	. . . . . . 7 class 𝑓
8	5, 5, 7	wf 6532	. . . . . 6 wff 𝑓:𝒫 𝑥⟶𝒫 𝑥
9		vs	. . . . . . . . 9 setvar 𝑠
10		vc	. . . . . . . . 9 setvar 𝑐
11	9, 10	wel 2143	. . . . . . . 8 wff 𝑠 ∈ 𝑐
12	9	cv 1568	. . . . . . . . . . . . 13 class 𝑠
13	12	cpw 4561	. . . . . . . . . . . 12 class 𝒫 𝑠
14		cfn 8941	. . . . . . . . . . . 12 class Fin
15	13, 14	cin 3903	. . . . . . . . . . 11 class (𝒫 𝑠 ∩ Fin)
16	7, 15	cima 5663	. . . . . . . . . 10 class (𝑓 “ (𝒫 𝑠 ∩ Fin))
17	16	cuni 4871	. . . . . . . . 9 class ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin))
18	17, 12	wss 3904	. . . . . . . 8 wff ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠
19	11, 18	wb 209	. . . . . . 7 wff (𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
20	19, 9, 5	wral 3078	. . . . . 6 wff ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
21	8, 20	wa 400	. . . . 5 wff (𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
22	21, 6	wex 1808	. . . 4 wff ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
23		cmre 17640	. . . . 5 class Moore
24	4, 23	cfv 6536	. . . 4 class (Moore‘𝑥)
25	22, 10, 24	crab 3415	. . 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 1569	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 used by:  isacs  17713