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Definition df-acs 17533
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 9638 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 = (π‘₯ ∈ V ↦ {𝑐 ∈ (Mooreβ€˜π‘₯) ∣ βˆƒπ‘“(𝑓:𝒫 π‘₯βŸΆπ’« π‘₯ ∧ βˆ€π‘  ∈ 𝒫 π‘₯(𝑠 ∈ 𝑐 ↔ βˆͺ (𝑓 β€œ (𝒫 𝑠 ∩ Fin)) βŠ† 𝑠))})
Distinct variable group:   𝑓,𝑐,𝑠,π‘₯

Detailed syntax breakdown of Definition df-acs
StepHypRef Expression
1 cacs 17529 . 2 class ACS
2 vx . . 3 setvar π‘₯
3 cvv 3475 . . 3 class V
42cv 1541 . . . . . . . 8 class π‘₯
54cpw 4603 . . . . . . 7 class 𝒫 π‘₯
6 vf . . . . . . . 8 setvar 𝑓
76cv 1541 . . . . . . 7 class 𝑓
85, 5, 7wf 6540 . . . . . 6 wff 𝑓:𝒫 π‘₯βŸΆπ’« π‘₯
9 vs . . . . . . . . 9 setvar 𝑠
10 vc . . . . . . . . 9 setvar 𝑐
119, 10wel 2108 . . . . . . . 8 wff 𝑠 ∈ 𝑐
129cv 1541 . . . . . . . . . . . . 13 class 𝑠
1312cpw 4603 . . . . . . . . . . . 12 class 𝒫 𝑠
14 cfn 8939 . . . . . . . . . . . 12 class Fin
1513, 14cin 3948 . . . . . . . . . . 11 class (𝒫 𝑠 ∩ Fin)
167, 15cima 5680 . . . . . . . . . 10 class (𝑓 β€œ (𝒫 𝑠 ∩ Fin))
1716cuni 4909 . . . . . . . . 9 class βˆͺ (𝑓 β€œ (𝒫 𝑠 ∩ Fin))
1817, 12wss 3949 . . . . . . . 8 wff βˆͺ (𝑓 β€œ (𝒫 𝑠 ∩ Fin)) βŠ† 𝑠
1911, 18wb 205 . . . . . . 7 wff (𝑠 ∈ 𝑐 ↔ βˆͺ (𝑓 β€œ (𝒫 𝑠 ∩ Fin)) βŠ† 𝑠)
2019, 9, 5wral 3062 . . . . . 6 wff βˆ€π‘  ∈ 𝒫 π‘₯(𝑠 ∈ 𝑐 ↔ βˆͺ (𝑓 β€œ (𝒫 𝑠 ∩ Fin)) βŠ† 𝑠)
218, 20wa 397 . . . . 5 wff (𝑓:𝒫 π‘₯βŸΆπ’« π‘₯ ∧ βˆ€π‘  ∈ 𝒫 π‘₯(𝑠 ∈ 𝑐 ↔ βˆͺ (𝑓 β€œ (𝒫 𝑠 ∩ Fin)) βŠ† 𝑠))
2221, 6wex 1782 . . . 4 wff βˆƒπ‘“(𝑓:𝒫 π‘₯βŸΆπ’« π‘₯ ∧ βˆ€π‘  ∈ 𝒫 π‘₯(𝑠 ∈ 𝑐 ↔ βˆͺ (𝑓 β€œ (𝒫 𝑠 ∩ Fin)) βŠ† 𝑠))
23 cmre 17526 . . . . 5 class Moore
244, 23cfv 6544 . . . 4 class (Mooreβ€˜π‘₯)
2522, 10, 24crab 3433 . . 3 class {𝑐 ∈ (Mooreβ€˜π‘₯) ∣ βˆƒπ‘“(𝑓:𝒫 π‘₯βŸΆπ’« π‘₯ ∧ βˆ€π‘  ∈ 𝒫 π‘₯(𝑠 ∈ 𝑐 ↔ βˆͺ (𝑓 β€œ (𝒫 𝑠 ∩ Fin)) βŠ† 𝑠))}
262, 3, 25cmpt 5232 . 2 class (π‘₯ ∈ V ↦ {𝑐 ∈ (Mooreβ€˜π‘₯) ∣ βˆƒπ‘“(𝑓:𝒫 π‘₯βŸΆπ’« π‘₯ ∧ βˆ€π‘  ∈ 𝒫 π‘₯(𝑠 ∈ 𝑐 ↔ βˆͺ (𝑓 β€œ (𝒫 𝑠 ∩ Fin)) βŠ† 𝑠))})
271, 26wceq 1542 1 wff ACS = (π‘₯ ∈ V ↦ {𝑐 ∈ (Mooreβ€˜π‘₯) ∣ βˆƒπ‘“(𝑓:𝒫 π‘₯βŸΆπ’« π‘₯ ∧ βˆ€π‘  ∈ 𝒫 π‘₯(𝑠 ∈ 𝑐 ↔ βˆͺ (𝑓 β€œ (𝒫 𝑠 ∩ Fin)) βŠ† 𝑠))})
Colors of variables: wff setvar class
This definition is referenced by:  isacs  17595
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