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Definition df-acs 16850
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 9095 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 16846 . 2 class ACS
2 vx . . 3 setvar 𝑥
3 cvv 3495 . . 3 class V
42cv 1527 . . . . . . . 8 class 𝑥
54cpw 4537 . . . . . . 7 class 𝒫 𝑥
6 vf . . . . . . . 8 setvar 𝑓
76cv 1527 . . . . . . 7 class 𝑓
85, 5, 7wf 6345 . . . . . 6 wff 𝑓:𝒫 𝑥⟶𝒫 𝑥
9 vs . . . . . . . . 9 setvar 𝑠
10 vc . . . . . . . . 9 setvar 𝑐
119, 10wel 2106 . . . . . . . 8 wff 𝑠𝑐
129cv 1527 . . . . . . . . . . . . 13 class 𝑠
1312cpw 4537 . . . . . . . . . . . 12 class 𝒫 𝑠
14 cfn 8498 . . . . . . . . . . . 12 class Fin
1513, 14cin 3934 . . . . . . . . . . 11 class (𝒫 𝑠 ∩ Fin)
167, 15cima 5552 . . . . . . . . . 10 class (𝑓 “ (𝒫 𝑠 ∩ Fin))
1716cuni 4832 . . . . . . . . 9 class (𝑓 “ (𝒫 𝑠 ∩ Fin))
1817, 12wss 3935 . . . . . . . 8 wff (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠
1911, 18wb 207 . . . . . . 7 wff (𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
2019, 9, 5wral 3138 . . . . . 6 wff 𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
218, 20wa 396 . . . . 5 wff (𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
2221, 6wex 1771 . . . 4 wff 𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
23 cmre 16843 . . . . 5 class Moore
244, 23cfv 6349 . . . 4 class (Moore‘𝑥)
2522, 10, 24crab 3142 . . 3 class {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))}
262, 3, 25cmpt 5138 . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
271, 26wceq 1528 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  16912
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