Theorem isnacs2 39310

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.

Step	Hyp	Ref	Expression
1		isnacs.f	. . 3 ⊢ 𝐹 = (mrCls‘𝐶)
2	1	isnacs 39308	. 2 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))
3		acsmre 16925	. . . . . . . . 9 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
4	1	mrcf 16882	. . . . . . . . 9 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
5		ffn 6516	. . . . . . . . 9 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → 𝐹 Fn 𝒫 𝑋)
6	3, 4, 5	3syl 18	. . . . . . . 8 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐹 Fn 𝒫 𝑋)
7		inss1 4207	. . . . . . . 8 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
8		fvelimab 6739	. . . . . . . 8 ⊢ ((𝐹 Fn 𝒫 𝑋 ∧ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋) → (𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑔) = 𝑠))
9	6, 7, 8	sylancl 588	. . . . . . 7 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑔) = 𝑠))
10		eqcom 2830	. . . . . . . 8 ⊢ (𝑠 = (𝐹‘𝑔) ↔ (𝐹‘𝑔) = 𝑠)
11	10	rexbii 3249	. . . . . . 7 ⊢ (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑔) = 𝑠)
12	9, 11	syl6rbbr 292	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
13	12	ralbidv 3199	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ ∀𝑠 ∈ 𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
14		dfss3 3958	. . . . 5 ⊢ (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∀𝑠 ∈ 𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)))
15	13, 14	syl6bbr 291	. . . 4 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ 𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
16		imassrn 5942	. . . . . . 7 ⊢ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ ran 𝐹
17		frn 6522	. . . . . . . 8 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → ran 𝐹 ⊆ 𝐶)
18	3, 4, 17	3syl 18	. . . . . . 7 ⊢ (𝐶 ∈ (ACS‘𝑋) → ran 𝐹 ⊆ 𝐶)
19	16, 18	sstrid 3980	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶)
20	19	biantrurd 535	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶 ∧ 𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)))))
21		eqss 3984	. . . . 5 ⊢ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶 ↔ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶 ∧ 𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
22	20, 21	syl6bbr 291	. . . 4 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
23	15, 22	bitrd 281	. . 3 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
24	23	pm5.32i 577	. 2 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
25	2, 24	bitri 277	1 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))

Syntax hints:   ↔ wb 208   ∧ wa 398   = wceq 1537   ∈ wcel 2114  ∀wral 3140  ∃wrex 3141   ∩ cin 3937   ⊆ wss 3938  𝒫 cpw 4541  ran crn 5558   “ cima 5560   Fn wfn 6352  ⟶wf 6353  ‘cfv 6357  Fincfn 8511  Moorecmre 16855  mrClscmrc 16856  ACScacs 16858  NoeACScnacs 39306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pow 5268  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-int 4879  df-br 5069  df-opab 5131  df-mpt 5149  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570  df-iota 6316  df-fun 6359  df-fn 6360  df-f 6361  df-fv 6365  df-mre 16859  df-mrc 16860  df-acs 16862  df-nacs 39307

This theorem is referenced by:  nacsacs  39313