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Theorem isnacs2 36784
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 36782 . 2 (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔)))
3 acsmre 16245 . . . . . . . . 9 (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
41mrcf 16201 . . . . . . . . 9 (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋𝐶)
5 ffn 6007 . . . . . . . . 9 (𝐹:𝒫 𝑋𝐶𝐹 Fn 𝒫 𝑋)
63, 4, 53syl 18 . . . . . . . 8 (𝐶 ∈ (ACS‘𝑋) → 𝐹 Fn 𝒫 𝑋)
7 inss1 3816 . . . . . . . 8 (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
8 fvelimab 6215 . . . . . . . 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 2628 . . . . . . . 8 (𝑠 = (𝐹𝑔) ↔ (𝐹𝑔) = 𝑠)
1110rexbii 3035 . . . . . . 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 2981 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → (∀𝑠𝐶𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹𝑔) ↔ ∀𝑠𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
14 dfss3 3577 . . . . 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 5441 . . . . . . 7 (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ ran 𝐹
17 frn 6015 . . . . . . . 8 (𝐹:𝒫 𝑋𝐶 → ran 𝐹𝐶)
183, 4, 173syl 18 . . . . . . 7 (𝐶 ∈ (ACS‘𝑋) → ran 𝐹𝐶)
1916, 18syl5ss 3598 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) → (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶)
2019biantrurd 529 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)))))
21 eqss 3602 . . . . 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 668 . 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 384   = wceq 1480  wcel 1987  wral 2907  wrex 2908  cin 3558  wss 3559  𝒫 cpw 4135  ran crn 5080  cima 5082   Fn wfn 5847  wf 5848  cfv 5852  Fincfn 7907  Moorecmre 16174  mrClscmrc 16175  ACScacs 16177  NoeACScnacs 36780
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3191  df-sbc 3422  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-int 4446  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-fv 5860  df-mre 16178  df-mrc 16179  df-acs 16181  df-nacs 36781
This theorem is referenced by:  nacsacs  36787
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