MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-acs Structured version   Visualization version   GIF version

Definition df-acs 16731
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 8899 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 16727 . 2 class ACS
2 vx . . 3 setvar 𝑥
3 cvv 3410 . . 3 class V
42cv 1507 . . . . . . . 8 class 𝑥
54cpw 4417 . . . . . . 7 class 𝒫 𝑥
6 vf . . . . . . . 8 setvar 𝑓
76cv 1507 . . . . . . 7 class 𝑓
85, 5, 7wf 6182 . . . . . 6 wff 𝑓:𝒫 𝑥⟶𝒫 𝑥
9 vs . . . . . . . . 9 setvar 𝑠
10 vc . . . . . . . . 9 setvar 𝑐
119, 10wel 2052 . . . . . . . 8 wff 𝑠𝑐
129cv 1507 . . . . . . . . . . . . 13 class 𝑠
1312cpw 4417 . . . . . . . . . . . 12 class 𝒫 𝑠
14 cfn 8305 . . . . . . . . . . . 12 class Fin
1513, 14cin 3823 . . . . . . . . . . 11 class (𝒫 𝑠 ∩ Fin)
167, 15cima 5407 . . . . . . . . . 10 class (𝑓 “ (𝒫 𝑠 ∩ Fin))
1716cuni 4709 . . . . . . . . 9 class (𝑓 “ (𝒫 𝑠 ∩ Fin))
1817, 12wss 3824 . . . . . . . 8 wff (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠
1911, 18wb 198 . . . . . . 7 wff (𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
2019, 9, 5wral 3083 . . . . . 6 wff 𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)
218, 20wa 387 . . . . 5 wff (𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
2221, 6wex 1743 . . . 4 wff 𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))
23 cmre 16724 . . . . 5 class Moore
244, 23cfv 6186 . . . 4 class (Moore‘𝑥)
2522, 10, 24crab 3087 . . 3 class {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))}
262, 3, 25cmpt 5005 . 2 class (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
271, 26wceq 1508 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  16793
  Copyright terms: Public domain W3C validator