Theorem acsfiel 17597

Description: A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 2-Apr-2015.)

Hypothesis

Ref	Expression
isacs2.f	⊢ 𝐹 = (mrCls‘𝐶)

Assertion

Ref	Expression
acsfiel	⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝐶 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆)))

Distinct variable groups:   𝑦,𝐶   𝑦,𝐹   𝑦,𝑆   𝑦,𝑋

Proof of Theorem acsfiel

Dummy variable 𝑠 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		acsmre 17595	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
2		mress 17536	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
3	1, 2	sylan 580	. . . 4 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
4	3	ex 413	. . 3 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑋))
5	4	pm4.71rd 563	. 2 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝐶 ↔ (𝑆 ⊆ 𝑋 ∧ 𝑆 ∈ 𝐶)))
6		eleq1 2821	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝐶 ↔ 𝑆 ∈ 𝐶))
7		pweq 4616	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
8	7	ineq1d 4211	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
9		sseq2 4008	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝐹‘𝑦) ⊆ 𝑠 ↔ (𝐹‘𝑦) ⊆ 𝑆))
10	8, 9	raleqbidv 3342	. . . . 5 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆))
11	6, 10	bibi12d 345	. . . 4 ⊢ (𝑠 = 𝑆 → ((𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠) ↔ (𝑆 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆)))
12		isacs2.f	. . . . . . 7 ⊢ 𝐹 = (mrCls‘𝐶)
13	12	isacs2 17596	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))
14	13	simprbi 497	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))
15	14	adantr 481	. . . 4 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))
16		elfvdm 6928	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ dom ACS)
17		elpw2g 5344	. . . . . 6 ⊢ (𝑋 ∈ dom ACS → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
18	16, 17	syl 17	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
19	18	biimpar 478	. . . 4 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋)
20	11, 15, 19	rspcdva 3613	. . 3 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆))
21	20	pm5.32da 579	. 2 ⊢ (𝐶 ∈ (ACS‘𝑋) → ((𝑆 ⊆ 𝑋 ∧ 𝑆 ∈ 𝐶) ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆)))
22	5, 21	bitrd 278	1 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝐶 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  ∀wral 3061   ∩ cin 3947   ⊆ wss 3948  𝒫 cpw 4602  dom cdm 5676  ‘cfv 6543  Fincfn 8938  Moorecmre 17525  mrClscmrc 17526  ACScacs 17528

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-mre 17529  df-mrc 17530  df-acs 17532

This theorem is referenced by:  acsfiel2  17598  isacs3lem  18494