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Definition df-acs 16457
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 8790 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 16453 . 2 class ACS
2 vx . . 3 setvar 𝑥
3 cvv 3398 . . 3 class V
42cv 1636 . . . . . . . 8 class 𝑥
54cpw 4358 . . . . . . 7 class 𝒫 𝑥
6 vf . . . . . . . 8 setvar 𝑓
76cv 1636 . . . . . . 7 class 𝑓
85, 5, 7wf 6100 . . . . . 6 wff 𝑓:𝒫 𝑥⟶𝒫 𝑥
9 vs . . . . . . . . 9 setvar 𝑠
10 vc . . . . . . . . 9 setvar 𝑐
119, 10wel 2158 . . . . . . . 8 wff 𝑠𝑐
129cv 1636 . . . . . . . . . . . . 13 class 𝑠
1312cpw 4358 . . . . . . . . . . . 12 class 𝒫 𝑠
14 cfn 8195 . . . . . . . . . . . 12 class Fin
1513, 14cin 3775 . . . . . . . . . . 11 class (𝒫 𝑠 ∩ Fin)
167, 15cima 5321 . . . . . . . . . 10 class (𝑓 “ (𝒫 𝑠 ∩ Fin))
1716cuni 4637 . . . . . . . . 9 class (𝑓 “ (𝒫 𝑠 ∩ Fin))
1817, 12wss 3776 . . . . . . . 8 wff (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠
1911, 18wb 197 . . . . . . 7 wff (𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
2019, 9, 5wral 3103 . . . . . 6 wff 𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
218, 20wa 384 . . . . 5 wff (𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
2221, 6wex 1859 . . . 4 wff 𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
23 cmre 16450 . . . . 5 class Moore
244, 23cfv 6104 . . . 4 class (Moore‘𝑥)
2522, 10, 24crab 3107 . . 3 class {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))}
262, 3, 25cmpt 4930 . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
271, 26wceq 1637 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  16519
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