Theorem acsfiel 17705

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 17703	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
2		mress 17640	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
3	1, 2	sylan 591	. . . 4 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
4	3	ex 417	. . 3 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑋))
5	4	pm4.71rd 571	. 2 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝐶 ↔ (𝑆 ⊆ 𝑋 ∧ 𝑆 ∈ 𝐶)))
6		eleq1 2851	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝐶 ↔ 𝑆 ∈ 𝐶))
7		pweq 4576	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
8	7	ineq1d 4172	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
9		sseq2 3963	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝐹‘𝑦) ⊆ 𝑠 ↔ (𝐹‘𝑦) ⊆ 𝑆))
10	8, 9	raleqbidv 3338	. . . . 5 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆))
11	6, 10	bibi12d 348	. . . 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 17704	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))
14	13	simprbi 502	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))
15	14	adantr 485	. . . 4 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))
16		elfvdm 6915	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ dom ACS)
17		elpw2g 5304	. . . . . 6 ⊢ (𝑋 ∈ dom ACS → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
18	16, 17	syl 18	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 ⊆ 𝑋))
19	18	biimpar 482	. . . 4 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋)
20	11, 15, 19	rspcdva 3582	. . 3 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆))
21	20	pm5.32da 589	. 2 ⊢ (𝐶 ∈ (ACS‘𝑋) → ((𝑆 ⊆ 𝑋 ∧ 𝑆 ∈ 𝐶) ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆)))
22	5, 21	bitrd 282	1 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝐶 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∀wral 3079   ∩ cin 3904   ⊆ wss 3905  𝒫 cpw 4562  dom cdm 5661  ‘cfv 6536  Fincfn 8939  Moorecmre 17629  mrClscmrc 17630  ACScacs 17632

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-mre 17633  df-mrc 17634  df-acs 17636

This theorem is referenced by:  acsfiel2  17706  isacs3lem  18593