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Theorem isacs 16770
Description: A set is an algebraic closure system iff it is specified by some function of the finite subsets, such that a set is closed iff it does not expand under the operation. (Contributed by Stefan O'Rear, 2-Apr-2015.)
Assertion
Ref Expression
isacs (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))))
Distinct variable groups:   𝐶,𝑓,𝑠   𝑓,𝑋,𝑠

Proof of Theorem isacs
Dummy variables 𝑐 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6527 . 2 (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ V)
2 elfvex 6527 . . 3 (𝐶 ∈ (Moore‘𝑋) → 𝑋 ∈ V)
32adantr 473 . 2 ((𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))) → 𝑋 ∈ V)
4 fveq2 6493 . . . . . 6 (𝑥 = 𝑋 → (Moore‘𝑥) = (Moore‘𝑋))
5 pweq 4419 . . . . . . . . 9 (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
65, 5feq23d 6333 . . . . . . . 8 (𝑥 = 𝑋 → (𝑓:𝒫 𝑥⟶𝒫 𝑥𝑓:𝒫 𝑋⟶𝒫 𝑋))
75raleqdv 3349 . . . . . . . 8 (𝑥 = 𝑋 → (∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠) ↔ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))
86, 7anbi12d 621 . . . . . . 7 (𝑥 = 𝑋 → ((𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)) ↔ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))))
98exbidv 1880 . . . . . 6 (𝑥 = 𝑋 → (∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)) ↔ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))))
104, 9rabeqbidv 3402 . . . . 5 (𝑥 = 𝑋 → {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))} = {𝑐 ∈ (Moore‘𝑋) ∣ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
11 df-acs 16708 . . . . 5 ACS = (𝑥 ∈ V ↦ {𝑐 ∈ (Moore‘𝑥) ∣ ∃𝑓(𝑓:𝒫 𝑥⟶𝒫 𝑥 ∧ ∀𝑠 ∈ 𝒫 𝑥(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
12 fvex 6506 . . . . . 6 (Moore‘𝑋) ∈ V
1312rabex 5085 . . . . 5 {𝑐 ∈ (Moore‘𝑋) ∣ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))} ∈ V
1410, 11, 13fvmpt 6589 . . . 4 (𝑋 ∈ V → (ACS‘𝑋) = {𝑐 ∈ (Moore‘𝑋) ∣ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))})
1514eleq2d 2845 . . 3 (𝑋 ∈ V → (𝐶 ∈ (ACS‘𝑋) ↔ 𝐶 ∈ {𝑐 ∈ (Moore‘𝑋) ∣ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))}))
16 eleq2 2848 . . . . . . . 8 (𝑐 = 𝐶 → (𝑠𝑐𝑠𝐶))
1716bibi1d 336 . . . . . . 7 (𝑐 = 𝐶 → ((𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠) ↔ (𝑠𝐶 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))
1817ralbidv 3141 . . . . . 6 (𝑐 = 𝐶 → (∀𝑠 ∈ 𝒫 𝑋(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠) ↔ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))
1918anbi2d 619 . . . . 5 (𝑐 = 𝐶 → ((𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)) ↔ (𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))))
2019exbidv 1880 . . . 4 (𝑐 = 𝐶 → (∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)) ↔ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))))
2120elrab 3589 . . 3 (𝐶 ∈ {𝑐 ∈ (Moore‘𝑋) ∣ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝑐 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))} ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))))
2215, 21syl6bb 279 . 2 (𝑋 ∈ V → (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠)))))
231, 3, 22pm5.21nii 371 1 (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∃𝑓(𝑓:𝒫 𝑋⟶𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 (𝑓 “ (𝒫 𝑠 ∩ Fin)) ⊆ 𝑠))))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387   = wceq 1507  wex 1742  wcel 2048  wral 3082  {crab 3086  Vcvv 3409  cin 3824  wss 3825  𝒫 cpw 4416   cuni 4706  cima 5403  wf 6178  cfv 6182  Fincfn 8298  Moorecmre 16701  ACScacs 16704
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-fv 6190  df-acs 16708
This theorem is referenced by:  acsmre  16771  isacs2  16772  isacs1i  16776  mreacs  16777
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