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Theorem isnacs 40563
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 6839 . 2 (𝐶 ∈ (NoeACS‘𝑋) → 𝑋 ∈ V)
2 elfvex 6839 . . 3 (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ V)
32adantr 482 . 2 ((𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)) → 𝑋 ∈ V)
4 fveq2 6804 . . . . . 6 (𝑥 = 𝑋 → (ACS‘𝑥) = (ACS‘𝑋))
5 pweq 4553 . . . . . . . . 9 (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
65ineq1d 4151 . . . . . . . 8 (𝑥 = 𝑋 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑋 ∩ Fin))
76rexeqdv 3361 . . . . . . 7 (𝑥 = 𝑋 → (∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)))
87ralbidv 3171 . . . . . 6 (𝑥 = 𝑋 → (∀𝑠𝑐𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)))
94, 8rabeqbidv 3427 . . . . 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 40562 . . . . 5 NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
11 fvex 6817 . . . . . 6 (ACS‘𝑋) ∈ V
1211rabex 5265 . . . . 5 {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)} ∈ V
139, 10, 12fvmpt 6907 . . . 4 (𝑋 ∈ V → (NoeACS‘𝑋) = {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
1413eleq2d 2822 . . 3 (𝑋 ∈ V → (𝐶 ∈ (NoeACS‘𝑋) ↔ 𝐶 ∈ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)}))
15 fveq2 6804 . . . . . . . . 9 (𝑐 = 𝐶 → (mrCls‘𝑐) = (mrCls‘𝐶))
16 isnacs.f . . . . . . . . 9 𝐹 = (mrCls‘𝐶)
1715, 16eqtr4di 2794 . . . . . . . 8 (𝑐 = 𝐶 → (mrCls‘𝑐) = 𝐹)
1817fveq1d 6806 . . . . . . 7 (𝑐 = 𝐶 → ((mrCls‘𝑐)‘𝑔) = (𝐹𝑔))
1918eqeq2d 2747 . . . . . 6 (𝑐 = 𝐶 → (𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ 𝑠 = (𝐹𝑔)))
2019rexbidv 3172 . . . . 5 (𝑐 = 𝐶 → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
2120raleqbi1dv 3352 . . . 4 (𝑐 = 𝐶 → (∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
2221elrab 3629 . . 3 (𝐶 ∈ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)} ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
2314, 22bitrdi 287 . 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 205  wa 397   = wceq 1539  wcel 2104  wral 3062  wrex 3071  {crab 3284  Vcvv 3437  cin 3891  𝒫 cpw 4539  cfv 6458  Fincfn 8764  mrClscmrc 17337  ACScacs 17339  NoeACScnacs 40561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2707  ax-sep 5232  ax-nul 5239  ax-pr 5361
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3287  df-v 3439  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4566  df-pr 4568  df-op 4572  df-uni 4845  df-br 5082  df-opab 5144  df-mpt 5165  df-id 5500  df-xp 5606  df-rel 5607  df-cnv 5608  df-co 5609  df-dm 5610  df-iota 6410  df-fun 6460  df-fv 6466  df-nacs 40562
This theorem is referenced by:  nacsfg  40564  isnacs2  40565  isnacs3  40569  islnr3  40978
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