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Theorem isnacs2 42717
Description: Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypothesis
Ref Expression
isnacs.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
isnacs2 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))

Proof of Theorem isnacs2
Dummy variables 𝑔 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnacs.f . . 3 𝐹 = (mrCls‘𝐶)
21isnacs 42715 . 2 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
3 eqcom 2744 . . . . . . . 8 (𝑠 = (𝐹𝑔) ↔ (𝐹𝑔) = 𝑠)
43rexbii 3094 . . . . . . 7 (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑔) = 𝑠)
5 acsmre 17695 . . . . . . . . 9 (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
61mrcf 17652 . . . . . . . . 9 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
7 ffn 6736 . . . . . . . . 9 (𝐹:𝒫 𝑋𝐶𝐹 Fn 𝒫 𝑋)
85, 6, 73syl 18 . . . . . . . 8 (𝐶 ∈ (ACS‘𝑋) → 𝐹 Fn 𝒫 𝑋)
9 inss1 4237 . . . . . . . 8 (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
10 fvelimab 6981 . . . . . . . 8 ((𝐹 Fn 𝒫 𝑋 ∧ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋) → (𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑔) = 𝑠))
118, 9, 10sylancl 586 . . . . . . 7 (𝐶 ∈ (ACS‘𝑋) → (𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑔) = 𝑠))
124, 11bitr4id 290 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
1312ralbidv 3178 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → (∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ ∀𝑠𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
14 dfss3 3972 . . . . 5 (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∀𝑠𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)))
1513, 14bitr4di 289 . . . 4 (𝐶 ∈ (ACS‘𝑋) → (∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ 𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
16 imassrn 6089 . . . . . . 7 (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ ran 𝐹
17 frn 6743 . . . . . . . 8 (𝐹:𝒫 𝑋𝐶 → ran 𝐹𝐶)
185, 6, 173syl 18 . . . . . . 7 (𝐶 ∈ (ACS‘𝑋) → ran 𝐹𝐶)
1916, 18sstrid 3995 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) → (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶)
2019biantrurd 532 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)))))
21 eqss 3999 . . . . 5 ((𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶 ↔ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
2220, 21bitr4di 289 . . . 4 (𝐶 ∈ (ACS‘𝑋) → (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
2315, 22bitrd 279 . . 3 (𝐶 ∈ (ACS‘𝑋) → (∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
2423pm5.32i 574 . 2 ((𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
252, 24bitri 275 1 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wcel 2108  wral 3061  wrex 3070  cin 3950  wss 3951  𝒫 cpw 4600  ran crn 5686  cima 5688   Fn wfn 6556  wf 6557  cfv 6561  Fincfn 8985  Moorecmre 17625  mrClscmrc 17626  ACScacs 17628  NoeACScnacs 42713
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 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-int 4947  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17629  df-mrc 17630  df-acs 17632  df-nacs 42714
This theorem is referenced by:  nacsacs  42720
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