Theorem isnacs2 43670

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 43668	. 2 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))
3		eqcom 2768	. . . . . . . 8 ⊢ (𝑠 = (𝐹‘𝑔) ↔ (𝐹‘𝑔) = 𝑠)
4	3	rexbii 3110	. . . . . . 7 ⊢ (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑔) = 𝑠)
5		acsmre 17806	. . . . . . . . 9 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
6	1	mrcf 17763	. . . . . . . . 9 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
7		ffn 6701	. . . . . . . . 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 4182	. . . . . . . 8 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
10		fvelimab 6949	. . . . . . . 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 3186	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ ∀𝑠 ∈ 𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
14		dfss3 3920	. . . . 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 6065	. . . . . . 7 ⊢ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ ran 𝐹
17		frn 6709	. . . . . . . 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 3942	. . . . . 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 3946	. . . . 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 2145  ∀wral 3077  ∃wrex 3087   ∩ cin 3898   ⊆ wss 3899  𝒫 cpw 4557  ran crn 5652   “ cima 5654   Fn wfn 6526  ⟶wf 6527  ‘cfv 6531  Fincfn 8957  Moorecmre 17732  mrClscmrc 17733  ACScacs 17735  NoeACScnacs 43666

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7740

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 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-iota 6487  df-fun 6533  df-fn 6534  df-f 6535  df-fv 6539  df-mre 17736  df-mrc 17737  df-acs 17739  df-nacs 43667

This theorem is used by:  nacsacs  43673