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Theorem acsfiel 16917
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 16915 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 mress 16856 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
31, 2sylan 583 . . . 4 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
43ex 416 . . 3 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶𝑆𝑋))
54pm4.71rd 566 . 2 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶 ↔ (𝑆𝑋𝑆𝐶)))
6 eleq1 2877 . . . . 5 (𝑠 = 𝑆 → (𝑠𝐶𝑆𝐶))
7 pweq 4513 . . . . . . 7 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
87ineq1d 4138 . . . . . 6 (𝑠 = 𝑆 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
9 sseq2 3941 . . . . . 6 (𝑠 = 𝑆 → ((𝐹𝑦) ⊆ 𝑠 ↔ (𝐹𝑦) ⊆ 𝑆))
108, 9raleqbidv 3354 . . . . 5 (𝑠 = 𝑆 → (∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))
116, 10bibi12d 349 . . . 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 16916 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))
1413simprbi 500 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))
1514adantr 484 . . . 4 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))
16 elfvdm 6677 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ dom ACS)
17 elpw2g 5211 . . . . . 6 (𝑋 ∈ dom ACS → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1816, 17syl 17 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1918biimpar 481 . . . 4 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
2011, 15, 19rspcdva 3573 . . 3 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))
2120pm5.32da 582 . 2 (𝐶 ∈ (ACS‘𝑋) → ((𝑆𝑋𝑆𝐶) ↔ (𝑆𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆)))
225, 21bitrd 282 1 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶 ↔ (𝑆𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2111  wral 3106  cin 3880  wss 3881  𝒫 cpw 4497  dom cdm 5519  cfv 6324  Fincfn 8492  Moorecmre 16845  mrClscmrc 16846  ACScacs 16848
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-fv 6332  df-mre 16849  df-mrc 16850  df-acs 16852
This theorem is referenced by:  acsfiel2  16918  isacs3lem  17768
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