Theorem isnacs2 42693

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 42691	. 2 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))
3		eqcom 2741	. . . . . . . 8 ⊢ (𝑠 = (𝐹‘𝑔) ↔ (𝐹‘𝑔) = 𝑠)
4	3	rexbii 3091	. . . . . . 7 ⊢ (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑔) = 𝑠)
5		acsmre 17696	. . . . . . . . 9 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
6	1	mrcf 17653	. . . . . . . . 9 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
7		ffn 6736	. . . . . . . . 9 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → 𝐹 Fn 𝒫 𝑋)
8	5, 6, 7	3syl 18	. . . . . . . 8 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐹 Fn 𝒫 𝑋)
9		inss1 4244	. . . . . . . 8 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
10		fvelimab 6980	. . . . . . . 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 586	. . . . . . 7 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑔) = 𝑠))
12	4, 11	bitr4id 290	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
13	12	ralbidv 3175	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ ∀𝑠 ∈ 𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
14		dfss3 3983	. . . . 5 ⊢ (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ∀𝑠 ∈ 𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin)))
15	13, 14	bitr4di 289	. . . 4 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ 𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
16		imassrn 6090	. . . . . . 7 ⊢ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ ran 𝐹
17		frn 6743	. . . . . . . 8 ⊢ (𝐹:𝒫 𝑋⟶𝐶 → ran 𝐹 ⊆ 𝐶)
18	5, 6, 17	3syl 18	. . . . . . 7 ⊢ (𝐶 ∈ (ACS‘𝑋) → ran 𝐹 ⊆ 𝐶)
19	16, 18	sstrid 4006	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶)
20	19	biantrurd 532	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶 ∧ 𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)))))
21		eqss 4010	. . . . 5 ⊢ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶 ↔ ((𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ 𝐶 ∧ 𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
22	20, 21	bitr4di 289	. . . 4 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝐶 ⊆ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ↔ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
23	15, 22	bitrd 279	. . 3 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))
24	23	pm5.32i 574	. 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 275	1 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ (𝐹 “ (𝒫 𝑋 ∩ Fin)) = 𝐶))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1536   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067   ∩ cin 3961   ⊆ wss 3962  𝒫 cpw 4604  ran crn 5689   “ cima 5691   Fn wfn 6557  ⟶wf 6558  ‘cfv 6562  Fincfn 8983  Moorecmre 17626  mrClscmrc 17627  ACScacs 17629  NoeACScnacs 42689

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570  df-mre 17630  df-mrc 17631  df-acs 17633  df-nacs 42690

This theorem is referenced by:  nacsacs  42696