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Theorem isnacs 39179
Description: Expand definition of Noetherian-type closure system. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypothesis
Ref Expression
isnacs.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
isnacs (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
Distinct variable groups:   𝐶,𝑔,𝑠   𝑔,𝐹,𝑠   𝑔,𝑋,𝑠

Proof of Theorem isnacs
Dummy variables 𝑐 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 6696 . 2 (𝐶 ∈ (NoeACS‘𝑋) → 𝑋 ∈ V)
2 elfvex 6696 . . 3 (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ V)
32adantr 481 . 2 ((𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)) → 𝑋 ∈ V)
4 fveq2 6663 . . . . . 6 (𝑥 = 𝑋 → (ACS‘𝑥) = (ACS‘𝑋))
5 pweq 4538 . . . . . . . . 9 (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
65ineq1d 4185 . . . . . . . 8 (𝑥 = 𝑋 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑋 ∩ Fin))
76rexeqdv 3414 . . . . . . 7 (𝑥 = 𝑋 → (∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)))
87ralbidv 3194 . . . . . 6 (𝑥 = 𝑋 → (∀𝑠𝑐𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)))
94, 8rabeqbidv 3483 . . . . 5 (𝑥 = 𝑋 → {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)} = {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
10 df-nacs 39178 . . . . 5 NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
11 fvex 6676 . . . . . 6 (ACS‘𝑋) ∈ V
1211rabex 5226 . . . . 5 {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)} ∈ V
139, 10, 12fvmpt 6761 . . . 4 (𝑋 ∈ V → (NoeACS‘𝑋) = {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
1413eleq2d 2895 . . 3 (𝑋 ∈ V → (𝐶 ∈ (NoeACS‘𝑋) ↔ 𝐶 ∈ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)}))
15 fveq2 6663 . . . . . . . . 9 (𝑐 = 𝐶 → (mrCls‘𝑐) = (mrCls‘𝐶))
16 isnacs.f . . . . . . . . 9 𝐹 = (mrCls‘𝐶)
1715, 16syl6eqr 2871 . . . . . . . 8 (𝑐 = 𝐶 → (mrCls‘𝑐) = 𝐹)
1817fveq1d 6665 . . . . . . 7 (𝑐 = 𝐶 → ((mrCls‘𝑐)‘𝑔) = (𝐹𝑔))
1918eqeq2d 2829 . . . . . 6 (𝑐 = 𝐶 → (𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ 𝑠 = (𝐹𝑔)))
2019rexbidv 3294 . . . . 5 (𝑐 = 𝐶 → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
2120raleqbi1dv 3401 . . . 4 (𝑐 = 𝐶 → (∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
2221elrab 3677 . . 3 (𝐶 ∈ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)} ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
2314, 22syl6bb 288 . 2 (𝑋 ∈ V → (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔))))
241, 3, 23pm5.21nii 380 1 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396   = wceq 1528  wcel 2105  wral 3135  wrex 3136  {crab 3139  Vcvv 3492  cin 3932  𝒫 cpw 4535  cfv 6348  Fincfn 8497  mrClscmrc 16842  ACScacs 16844  NoeACScnacs 39177
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-pw 4537  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-iota 6307  df-fun 6350  df-fv 6356  df-nacs 39178
This theorem is referenced by:  nacsfg  39180  isnacs2  39181  isnacs3  39185  islnr3  39593
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