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Theorem acsfiel 17699
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) → (𝑆𝐶 ↔ (𝑆𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 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.
StepHypRef Expression
1 acsmre 17697 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 mress 17638 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
31, 2sylan 580 . . . 4 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
43ex 412 . . 3 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶𝑆𝑋))
54pm4.71rd 562 . 2 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶 ↔ (𝑆𝑋𝑆𝐶)))
6 eleq1 2827 . . . . 5 (𝑠 = 𝑆 → (𝑠𝐶𝑆𝐶))
7 pweq 4619 . . . . . . 7 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
87ineq1d 4227 . . . . . 6 (𝑠 = 𝑆 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
9 sseq2 4022 . . . . . 6 (𝑠 = 𝑆 → ((𝐹𝑦) ⊆ 𝑠 ↔ (𝐹𝑦) ⊆ 𝑆))
108, 9raleqbidv 3344 . . . . 5 (𝑠 = 𝑆 → (∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))
116, 10bibi12d 345 . . . 4 (𝑠 = 𝑆 → ((𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠) ↔ (𝑆𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆)))
12 isacs2.f . . . . . . 7 𝐹 = (mrCls‘𝐶)
1312isacs2 17698 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))
1413simprbi 496 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))
1514adantr 480 . . . 4 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))
16 elfvdm 6944 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ dom ACS)
17 elpw2g 5339 . . . . . 6 (𝑋 ∈ dom ACS → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1816, 17syl 17 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1918biimpar 477 . . . 4 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
2011, 15, 19rspcdva 3623 . . 3 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))
2120pm5.32da 579 . 2 (𝐶 ∈ (ACS‘𝑋) → ((𝑆𝑋𝑆𝐶) ↔ (𝑆𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆)))
225, 21bitrd 279 1 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶 ↔ (𝑆𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  wral 3059  cin 3962  wss 3963  𝒫 cpw 4605  dom cdm 5689  cfv 6563  Fincfn 8984  Moorecmre 17627  mrClscmrc 17628  ACScacs 17630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-mre 17631  df-mrc 17632  df-acs 17634
This theorem is referenced by:  acsfiel2  17700  isacs3lem  18600
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