Definition df-acs 17512

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 9556 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 17508	. 2 class ACS
2		vx	. . 3 setvar 𝑥
3		cvv 3441	. . 3 class V
4	2	cv 1541	. . . . . . . 8 class 𝑥
5	4	cpw 4555	. . . . . . 7 class 𝒫 𝑥
6		vf	. . . . . . . 8 setvar 𝑓
7	6	cv 1541	. . . . . . 7 class 𝑓
8	5, 5, 7	wf 6489	. . . . . 6 wff 𝑓:𝒫 𝑥⟶𝒫 𝑥
9		vs	. . . . . . . . 9 setvar 𝑠
10		vc	. . . . . . . . 9 setvar 𝑐
11	9, 10	wel 2115	. . . . . . . 8 wff 𝑠 ∈ 𝑐
12	9	cv 1541	. . . . . . . . . . . . 13 class 𝑠
13	12	cpw 4555	. . . . . . . . . . . 12 class 𝒫 𝑠
14		cfn 8887	. . . . . . . . . . . 12 class Fin
15	13, 14	cin 3901	. . . . . . . . . . 11 class (𝒫 𝑠 ∩ Fin)
16	7, 15	cima 5628	. . . . . . . . . 10 class (𝑓 “ (𝒫 𝑠 ∩ Fin))
17	16	cuni 4864	. . . . . . . . 9 class ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin))
18	17, 12	wss 3902	. . . . . . . 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 3052	. . . . . 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 1781	. . . 4 wff ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠 ∈ 𝑐 ↔ ∪ (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
23		cmre 17505	. . . . 5 class Moore
24	4, 23	cfv 6493	. . . 4 class (Moore‘𝑥)
25	22, 10, 24	crab 3400	. . 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 5180	. 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 1542	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  17578