Theorem isnacs2 42948

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 42946	. 2 ⊢ (𝐶 ∈ (NoeACS‘𝑋) ↔ (𝐶 ∈ (ACS‘𝑋) ∧ ∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔)))
3		eqcom 2743	. . . . . . . 8 ⊢ (𝑠 = (𝐹‘𝑔) ↔ (𝐹‘𝑔) = 𝑠)
4	3	rexbii 3083	. . . . . . 7 ⊢ (∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)(𝐹‘𝑔) = 𝑠)
5		acsmre 17575	. . . . . . . . 9 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
6	1	mrcf 17532	. . . . . . . . 9 ⊢ (𝐶 ∈ (Moore‘𝑋) → 𝐹:𝒫 𝑋⟶𝐶)
7		ffn 6662	. . . . . . . . 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 4189	. . . . . . . 8 ⊢ (𝒫 𝑋 ∩ Fin) ⊆ 𝒫 𝑋
10		fvelimab 6906	. . . . . . . 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 3159	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → (∀𝑠 ∈ 𝐶 ∃𝑔 ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (𝐹‘𝑔) ↔ ∀𝑠 ∈ 𝐶 𝑠 ∈ (𝐹 “ (𝒫 𝑋 ∩ Fin))))
14		dfss3 3922	. . . . 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 6030	. . . . . . 7 ⊢ (𝐹 “ (𝒫 𝑋 ∩ Fin)) ⊆ ran 𝐹
17		frn 6669	. . . . . . . 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 3945	. . . . . 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 3949	. . . . 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 1541   ∈ wcel 2113  ∀wral 3051  ∃wrex 3060   ∩ cin 3900   ⊆ wss 3901  𝒫 cpw 4554  ran crn 5625   “ cima 5627   Fn wfn 6487  ⟶wf 6488  ‘cfv 6492  Fincfn 8883  Moorecmre 17501  mrClscmrc 17502  ACScacs 17504  NoeACScnacs 42944

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 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pow 5310  ax-pr 5377  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-pw 4556  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-int 4903  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fv 6500  df-mre 17505  df-mrc 17506  df-acs 17508  df-nacs 42945

This theorem is referenced by:  nacsacs  42951