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Theorem isnacs2 36287
 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 36285 . 2 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
3 acsmre 16136 . . . . . . . . 9 (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
41mrcf 16092 . . . . . . . . 9 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
5 ffn 5958 . . . . . . . . 9 (𝐹:𝒫 𝑋𝐶𝐹 Fn 𝒫 𝑋)
63, 4, 53syl 18 . . . . . . . 8 (𝐶 ∈ (ACS‘𝑋) → 𝐹 Fn 𝒫 𝑋)
7 inss1 3795 . . . . . . . 8 (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
8 fvelimab 6163 . . . . . . . 8 ((𝐹 Fn 𝒫 𝑋 ∧ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋) → (𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑔) = 𝑠))
96, 7, 8sylancl 693 . . . . . . 7 (𝐶 ∈ (ACS‘𝑋) → (𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑔) = 𝑠))
10 eqcom 2617 . . . . . . . 8 (𝑠 = (𝐹𝑔) ↔ (𝐹𝑔) = 𝑠)
1110rexbii 3023 . . . . . . 7 (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑔) = 𝑠)
129, 11syl6rbbr 278 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
1312ralbidv 2969 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → (∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ ∀𝑠𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
14 dfss3 3558 . . . . 5 (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∀𝑠𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)))
1513, 14syl6bbr 277 . . . 4 (𝐶 ∈ (ACS‘𝑋) → (∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ 𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
16 imassrn 5396 . . . . . . 7 (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ ran 𝐹
17 frn 5966 . . . . . . . 8 (𝐹:𝒫 𝑋𝐶 → ran 𝐹𝐶)
183, 4, 173syl 18 . . . . . . 7 (𝐶 ∈ (ACS‘𝑋) → ran 𝐹𝐶)
1916, 18syl5ss 3579 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) → (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶)
2019biantrurd 528 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)))))
21 eqss 3583 . . . . 5 ((𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶 ↔ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
2220, 21syl6bbr 277 . . . 4 (𝐶 ∈ (ACS‘𝑋) → (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
2315, 22bitrd 267 . . 3 (𝐶 ∈ (ACS‘𝑋) → (∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
2423pm5.32i 667 . 2 ((𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
252, 24bitri 263 1 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  ∀wral 2896  ∃wrex 2897   ∩ cin 3539   ⊆ wss 3540  𝒫 cpw 4108  ran crn 5039   “ cima 5041   Fn wfn 5799  ⟶wf 5800  ‘cfv 5804  Fincfn 7841  Moorecmre 16065  mrClscmrc 16066  ACScacs 16068  NoeACScnacs 36283 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-int 4411  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812  df-mre 16069  df-mrc 16070  df-acs 16072  df-nacs 36284 This theorem is referenced by:  nacsacs  36290
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