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Definition df-acs 16296
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 8578 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(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
Distinct variable group:   𝑓,𝑐,𝑠,𝑥

Detailed syntax breakdown of Definition df-acs
StepHypRef Expression
1 cacs 16292 . 2 class ACS
2 vx . . 3 setvar 𝑥
3 cvv 3231 . . 3 class V
42cv 1522 . . . . . . . 8 class 𝑥
54cpw 4191 . . . . . . 7 class 𝒫 𝑥
6 vf . . . . . . . 8 setvar 𝑓
76cv 1522 . . . . . . 7 class 𝑓
85, 5, 7wf 5922 . . . . . 6 wff 𝑓:𝒫 𝑥⟶𝒫 𝑥
9 vs . . . . . . . . 9 setvar 𝑠
10 vc . . . . . . . . 9 setvar 𝑐
119, 10wel 2031 . . . . . . . 8 wff 𝑠𝑐
129cv 1522 . . . . . . . . . . . . 13 class 𝑠
1312cpw 4191 . . . . . . . . . . . 12 class 𝒫 𝑠
14 cfn 7997 . . . . . . . . . . . 12 class Fin
1513, 14cin 3606 . . . . . . . . . . 11 class (𝒫 𝑠 ∩ Fin)
167, 15cima 5146 . . . . . . . . . 10 class (𝑓 “ (𝒫 𝑠 ∩ Fin))
1716cuni 4468 . . . . . . . . 9 class (𝑓 “ (𝒫 𝑠 ∩ Fin))
1817, 12wss 3607 . . . . . . . 8 wff (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠
1911, 18wb 196 . . . . . . 7 wff (𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
2019, 9, 5wral 2941 . . . . . 6 wff 𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
218, 20wa 383 . . . . 5 wff (𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
2221, 6wex 1744 . . . 4 wff 𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
23 cmre 16289 . . . . 5 class Moore
244, 23cfv 5926 . . . 4 class (Moore‘𝑥)
2522, 10, 24crab 2945 . . 3 class {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))}
262, 3, 25cmpt 4762 . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
271, 26wceq 1523 1 wff ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
Colors of variables: wff setvar class
This definition is referenced by:  isacs  16359
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