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Theorem nacsfg 41995
Description: In a Noetherian-type closure system, all closed sets are finitely generated. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypothesis
Ref Expression
isnacs.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
nacsfg ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑆𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔))
Distinct variable groups:   𝐶,𝑔   𝑔,𝐹   𝑆,𝑔   𝑔,𝑋

Proof of Theorem nacsfg
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 isnacs.f . . . . 5 𝐹 = (mrCls‘𝐶)
21isnacs 41994 . . . 4 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
32simprbi 496 . . 3 (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔))
4 eqeq1 2728 . . . . 5 (𝑠 = 𝑆 → (𝑠 = (𝐹𝑔) ↔ 𝑆 = (𝐹𝑔)))
54rexbidv 3170 . . . 4 (𝑠 = 𝑆 → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔)))
65rspcva 3602 . . 3 ((𝑆𝐶 ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔))
73, 6sylan2 592 . 2 ((𝑆𝐶𝐶 ∈ (NoeACS‘𝑋)) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔))
87ancoms 458 1 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑆𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1533  wcel 2098  wral 3053  wrex 3062  cin 3940  𝒫 cpw 4595  cfv 6534  Fincfn 8936  mrClscmrc 17532  ACScacs 17534  NoeACScnacs 41992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-mpt 5223  df-id 5565  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-iota 6486  df-fun 6536  df-fv 6542  df-nacs 41993
This theorem is referenced by:  isnacs3  42000
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