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Theorem isnacs 37761
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 6374 . 2 (𝐶 ∈ (NoeACS‘𝑋) → 𝑋 ∈ V)
2 elfvex 6374 . . 3 (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ V)
32adantr 472 . 2 ((𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)) → 𝑋 ∈ V)
4 fveq2 6344 . . . . . 6 (𝑥 = 𝑋 → (ACS‘𝑥) = (ACS‘𝑋))
5 pweq 4297 . . . . . . . . 9 (𝑥 = 𝑋 → 𝒫 𝑥 = 𝒫 𝑋)
65ineq1d 3948 . . . . . . . 8 (𝑥 = 𝑋 → (𝒫 𝑥 ∩ Fin) = (𝒫 𝑋 ∩ Fin))
76rexeqdv 3276 . . . . . . 7 (𝑥 = 𝑋 → (∃𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)))
87ralbidv 3116 . . . . . 6 (𝑥 = 𝑋 → (∀𝑠𝑐𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)))
94, 8rabeqbidv 3327 . . . . 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 37760 . . . . 5 NoeACS = (𝑥 ∈ V ↦ {𝑐 ∈ (ACS‘𝑥) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑥 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
11 fvex 6354 . . . . . 6 (ACS‘𝑋) ∈ V
1211rabex 4956 . . . . 5 {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)} ∈ V
139, 10, 12fvmpt 6436 . . . 4 (𝑋 ∈ V → (NoeACS‘𝑋) = {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)})
1413eleq2d 2817 . . 3 (𝑋 ∈ V → (𝐶 ∈ (NoeACS‘𝑋) ↔ 𝐶 ∈ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)}))
15 fveq2 6344 . . . . . . . . 9 (𝑐 = 𝐶 → (mrCls‘𝑐) = (mrCls‘𝐶))
16 isnacs.f . . . . . . . . 9 𝐹 = (mrCls‘𝐶)
1715, 16syl6eqr 2804 . . . . . . . 8 (𝑐 = 𝐶 → (mrCls‘𝑐) = 𝐹)
1817fveq1d 6346 . . . . . . 7 (𝑐 = 𝐶 → ((mrCls‘𝑐)‘𝑔) = (𝐹𝑔))
1918eqeq2d 2762 . . . . . 6 (𝑐 = 𝐶 → (𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ 𝑠 = (𝐹𝑔)))
2019rexbidv 3182 . . . . 5 (𝑐 = 𝐶 → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
2120raleqbi1dv 3277 . . . 4 (𝑐 = 𝐶 → (∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔) ↔ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
2221elrab 3496 . . 3 (𝐶 ∈ {𝑐 ∈ (ACS‘𝑋) ∣ ∀𝑠𝑐𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = ((mrCls‘𝑐)‘𝑔)} ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
2314, 22syl6bb 276 . 2 (𝑋 ∈ V → (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔))))
241, 3, 23pm5.21nii 367 1 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1624  wcel 2131  wral 3042  wrex 3043  {crab 3046  Vcvv 3332  cin 3706  𝒫 cpw 4294  cfv 6041  Fincfn 8113  mrClscmrc 16437  ACScacs 16439  NoeACScnacs 37759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1863  ax-4 1878  ax-5 1980  ax-6 2046  ax-7 2082  ax-8 2133  ax-9 2140  ax-10 2160  ax-11 2175  ax-12 2188  ax-13 2383  ax-ext 2732  ax-sep 4925  ax-nul 4933  ax-pow 4984  ax-pr 5047
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1627  df-ex 1846  df-nf 1851  df-sb 2039  df-eu 2603  df-mo 2604  df-clab 2739  df-cleq 2745  df-clel 2748  df-nfc 2883  df-ne 2925  df-ral 3047  df-rex 3048  df-rab 3051  df-v 3334  df-sbc 3569  df-dif 3710  df-un 3712  df-in 3714  df-ss 3721  df-nul 4051  df-if 4223  df-pw 4296  df-sn 4314  df-pr 4316  df-op 4320  df-uni 4581  df-br 4797  df-opab 4857  df-mpt 4874  df-id 5166  df-xp 5264  df-rel 5265  df-cnv 5266  df-co 5267  df-dm 5268  df-iota 6004  df-fun 6043  df-fv 6049  df-nacs 37760
This theorem is referenced by:  nacsfg  37762  isnacs2  37763  isnacs3  37767  islnr3  38179
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