Theorem acsfiel 17641

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 17639	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
2		mress 17580	. . . . 5 ⊢ ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
3	1, 2	sylan 578	. . . 4 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ∈ 𝐶) → 𝑆 ⊆ 𝑋)
4	3	ex 411	. . 3 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝐶 → 𝑆 ⊆ 𝑋))
5	4	pm4.71rd 561	. 2 ⊢ (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝐶 ↔ (𝑆 ⊆ 𝑋 ∧ 𝑆 ∈ 𝐶)))
6		eleq1 2817	. . . . 5 ⊢ (𝑠 = 𝑆 → (𝑠 ∈ 𝐶 ↔ 𝑆 ∈ 𝐶))
7		pweq 4620	. . . . . . 7 ⊢ (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
8	7	ineq1d 4213	. . . . . 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 3340	. . . . 5 ⊢ (𝑠 = 𝑆 → (∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆))
11	6, 10	bibi12d 344	. . . 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 17640	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠)))
14	13	simprbi 495	. . . . 5 ⊢ (𝐶 ∈ (ACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))
15	14	adantr 479	. . . 4 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑠))
16		elfvdm 6939	. . . . . 6 ⊢ (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ dom ACS)
17		elpw2g 5350	. . . . . 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 476	. . . 4 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → 𝑆 ∈ 𝒫 𝑋)
20	11, 15, 19	rspcdva 3612	. . 3 ⊢ ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆 ⊆ 𝑋) → (𝑆 ∈ 𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹‘𝑦) ⊆ 𝑆))
21	20	pm5.32da 577	. 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 394   = wceq 1533   ∈ wcel 2098  ∀wral 3058   ∩ cin 3948   ⊆ wss 3949  𝒫 cpw 4606  dom cdm 5682  ‘cfv 6553  Fincfn 8970  Moorecmre 17569  mrClscmrc 17570  ACScacs 17572

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pow 5369  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-int 4954  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-mre 17573  df-mrc 17574  df-acs 17576

This theorem is referenced by:  acsfiel2  17642  isacs3lem  18541