Theorem acsfiel 17634

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 17632	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
2		mress 17573	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
3	1, 2	sylan 579	. . . 4 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
4	3	ex 412	. . 3 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑋))
5	4	pm4.71rd 562	. 2 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝐶 ↔ (𝑆 ⊆ 𝑋 ∧ 𝑆 ∈ 𝐶)))
6		eleq1 2817	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝐶 ↔ 𝑆 ∈ 𝐶))
7		pweq 4617	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
8	7	ineq1d 4211	. . . . . 6 ⊢ (𝑠 = 𝑆 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
9		sseq2 4006	. . . . . 6 ⊢ (𝑠 = 𝑆 → ((𝐹‘𝑦) ⊆ 𝑠 ↔ (𝐹‘𝑦) ⊆ 𝑆))
10	8, 9	raleqbidv 3339	. . . . 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 17633	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))
14	13	simprbi 496	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))
15	14	adantr 480	. . . 4 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))
16		elfvdm 6934	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ dom ACS)
17		elpw2g 5346	. . . . . 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 477	. . . 4 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋)
20	11, 15, 19	rspcdva 3610	. . 3 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆))
21	20	pm5.32da 578	. 2 ⊢ (𝐶 ∈ (ACS‘𝑋) → ((𝑆 ⊆ 𝑋 ∧ 𝑆 ∈ 𝐶) ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆)))
22	5, 21	bitrd 279	1 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝐶 ↔ (𝑆 ⊆ 𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆)))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 395   = wceq 1534   ∈ wcel 2099  ∀wral 3058   ∩ cin 3946   ⊆ wss 3947  𝒫 cpw 4603  dom cdm 5678  ‘cfv 6548  Fincfn 8964  Moorecmre 17562  mrClscmrc 17563  ACScacs 17565

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-mre 17566  df-mrc 17567  df-acs 17569

This theorem is referenced by:  acsfiel2  17635  isacs3lem  18534