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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) → (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 17632 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 mress 17573 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
31, 2sylan 579 . . . 4 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
43ex 412 . . 3 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶𝑆𝑋))
54pm4.71rd 562 . 2 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶 ↔ (𝑆𝑋𝑆𝐶)))
6 eleq1 2817 . . . . 5 (𝑠 = 𝑆 → (𝑠𝐶𝑆𝐶))
7 pweq 4617 . . . . . . 7 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
87ineq1d 4211 . . . . . 6 (𝑠 = 𝑆 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
9 sseq2 4006 . . . . . 6 (𝑠 = 𝑆 → ((𝐹𝑦) ⊆ 𝑠 ↔ (𝐹𝑦) ⊆ 𝑆))
108, 9raleqbidv 3339 . . . . 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 17633 . . . . . 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 6934 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ dom ACS)
17 elpw2g 5346 . . . . . 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 3610 . . 3 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))
2120pm5.32da 578 . 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 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
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