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Theorem isnacs2 37763
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 37761 . 2 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
3 acsmre 16506 . . . . . . . . 9 (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
41mrcf 16463 . . . . . . . . 9 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
5 ffn 6198 . . . . . . . . 9 (𝐹:𝒫 𝑋𝐶𝐹 Fn 𝒫 𝑋)
63, 4, 53syl 18 . . . . . . . 8 (𝐶 ∈ (ACS‘𝑋) → 𝐹 Fn 𝒫 𝑋)
7 inss1 3968 . . . . . . . 8 (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
8 fvelimab 6407 . . . . . . . 8 ((𝐹 Fn 𝒫 𝑋 ∧ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋) → (𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑔) = 𝑠))
96, 7, 8sylancl 697 . . . . . . 7 (𝐶 ∈ (ACS‘𝑋) → (𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑔) = 𝑠))
10 eqcom 2759 . . . . . . . 8 (𝑠 = (𝐹𝑔) ↔ (𝐹𝑔) = 𝑠)
1110rexbii 3171 . . . . . . 7 (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹𝑔) = 𝑠)
129, 11syl6rbbr 279 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
1312ralbidv 3116 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → (∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ ∀𝑠𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
14 dfss3 3725 . . . . 5 (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∀𝑠𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)))
1513, 14syl6bbr 278 . . . 4 (𝐶 ∈ (ACS‘𝑋) → (∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ 𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
16 imassrn 5627 . . . . . . 7 (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ ran 𝐹
17 frn 6206 . . . . . . . 8 (𝐹:𝒫 𝑋𝐶 → ran 𝐹𝐶)
183, 4, 173syl 18 . . . . . . 7 (𝐶 ∈ (ACS‘𝑋) → ran 𝐹𝐶)
1916, 18syl5ss 3747 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) → (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶)
2019biantrurd 530 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)))))
21 eqss 3751 . . . . 5 ((𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶 ↔ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
2220, 21syl6bbr 278 . . . 4 (𝐶 ∈ (ACS‘𝑋) → (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
2315, 22bitrd 268 . . 3 (𝐶 ∈ (ACS‘𝑋) → (∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
2423pm5.32i 672 . 2 ((𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
252, 24bitri 264 1 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383   = wceq 1624  wcel 2131  wral 3042  wrex 3043  cin 3706  wss 3707  𝒫 cpw 4294  ran crn 5259  cima 5261   Fn wfn 6036  wf 6037  cfv 6041  Fincfn 8113  Moorecmre 16436  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  ax-un 7106
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-int 4620  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-rn 5269  df-res 5270  df-ima 5271  df-iota 6004  df-fun 6043  df-fn 6044  df-f 6045  df-fv 6049  df-mre 16440  df-mrc 16441  df-acs 16443  df-nacs 37760
This theorem is referenced by:  nacsacs  37766
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