Theorem isnacs2 43921

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 43919	. 2 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))
3		eqcom 2770	. . . . . . . 8 ⊢ (𝑠 = (𝐹‘𝑔) ↔ (𝐹‘𝑔) = 𝑠)
4	3	rexbii 3113	. . . . . . 7 ⊢ (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑔) = 𝑠)
5		acsmre 17858	. . . . . . . . 9 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
6	1	mrcf 17815	. . . . . . . . 9 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
7		ffn 6734	. . . . . . . . 9 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → 𝐹 Fn 𝒫 𝑋)
8	5, 6, 7	3syl 19	. . . . . . . 8 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐹 Fn 𝒫 𝑋)
9		inss1 4136	. . . . . . . 8 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
10		fvelimab 6983	. . . . . . . 8 ⊢ ((𝐹 Fn 𝒫 𝑋 ∧ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋) → (𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑔) = 𝑠))
11	8, 9, 10	sylancl 598	. . . . . . 7 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑔) = 𝑠))
12	4, 11	bitr4id 293	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
13	12	ralbidv 3189	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ ∀𝑠 ∈ 𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
14		dfss3 3902	. . . . 5 ⊢ (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∀𝑠 ∈ 𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)))
15	13, 14	bitr4di 292	. . . 4 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ 𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
16		imassrn 6211	. . . . . . 7 ⊢ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ ran 𝐹
17		frn 6742	. . . . . . . 8 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → ran 𝐹 ⊆ 𝐶)
18	5, 6, 17	3syl 19	. . . . . . 7 ⊢ (𝐶 ∈ (ACS‘𝑋) → ran 𝐹 ⊆ 𝐶)
19	16, 18	sstrid 3924	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶)
20	19	biantrurd 542	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶 ∧ 𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)))))
21		eqss 3928	. . . . 5 ⊢ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶 ↔ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶 ∧ 𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
22	20, 21	bitr4di 292	. . . 4 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
23	15, 22	bitrd 282	. . 3 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
24	23	pm5.32i 585	. 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 278	1 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2147  ∀wral 3080  ∃wrex 3090   ⊆ wss 3900   ∩ cin 4092  𝒫 cpw 4560  ran crn 5657   “ cima 5659   Fn wfn 6556  ⟶wf 6557  ‘cfv 6561  Fincfn 9004  Moorecmre 17784  mrClscmrc 17785  ACScacs 17787  NoeACScnacs 43917

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-nul 5264  ax-pow 5332  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-int 4911  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-ima 5669  df-iota 6515  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569  df-mre 17788  df-mrc 17789  df-acs 17791  df-nacs 43918

This theorem is used by:  nacsacs  43924