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Theorem nacsfg 41075
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 41074 . . . 4 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
32simprbi 498 . . 3 (𝐶 ∈ (NoeACS‘𝑋) → ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔))
4 eqeq1 2737 . . . . 5 (𝑠 = 𝑆 → (𝑠 = (𝐹𝑔) ↔ 𝑆 = (𝐹𝑔)))
54rexbidv 3172 . . . 4 (𝑠 = 𝑆 → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔)))
65rspcva 3581 . . 3 ((𝑆𝐶 ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔))
73, 6sylan2 594 . 2 ((𝑆𝐶𝐶 ∈ (NoeACS‘𝑋)) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔))
87ancoms 460 1 ((𝐶 ∈ (NoeACS‘𝑋) ∧ 𝑆𝐶) → ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (𝐹𝑔))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wral 3061  wrex 3070  cin 3913  𝒫 cpw 4564  cfv 6500  Fincfn 8889  mrClscmrc 17471  ACScacs 17473  NoeACScnacs 41072
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-nul 5267  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-iota 6452  df-fun 6502  df-fv 6508  df-nacs 41073
This theorem is referenced by:  isnacs3  41080
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