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Theorem acsfiel 16236
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 16234 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 mress 16174 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
31, 2sylan 488 . . . 4 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
43ex 450 . . 3 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶𝑆𝑋))
54pm4.71rd 666 . 2 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶 ↔ (𝑆𝑋𝑆𝐶)))
6 elfvdm 6177 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ dom ACS)
7 elpw2g 4787 . . . . . 6 (𝑋 ∈ dom ACS → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
86, 7syl 17 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
98biimpar 502 . . . 4 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
10 isacs2.f . . . . . . 7 𝐹 = (mrCls‘𝐶)
1110isacs2 16235 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))
1211simprbi 480 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))
1312adantr 481 . . . 4 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))
14 eleq1 2686 . . . . . 6 (𝑠 = 𝑆 → (𝑠𝐶𝑆𝐶))
15 pweq 4133 . . . . . . . 8 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
1615ineq1d 3791 . . . . . . 7 (𝑠 = 𝑆 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
17 sseq2 3606 . . . . . . 7 (𝑠 = 𝑆 → ((𝐹𝑦) ⊆ 𝑠 ↔ (𝐹𝑦) ⊆ 𝑆))
1816, 17raleqbidv 3141 . . . . . 6 (𝑠 = 𝑆 → (∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))
1914, 18bibi12d 335 . . . . 5 (𝑠 = 𝑆 → ((𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠) ↔ (𝑆𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆)))
2019rspcva 3293 . . . 4 ((𝑆 ∈ 𝒫 𝑋 ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)) → (𝑆𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))
219, 13, 20syl2anc 692 . . 3 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))
2221pm5.32da 672 . 2 (𝐶 ∈ (ACS‘𝑋) → ((𝑆𝑋𝑆𝐶) ↔ (𝑆𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆)))
235, 22bitrd 268 1 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶 ↔ (𝑆𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  cin 3554  wss 3555  𝒫 cpw 4130  dom cdm 5074  cfv 5847  Fincfn 7899  Moorecmre 16163  mrClscmrc 16164  ACScacs 16166
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fv 5855  df-mre 16167  df-mrc 16168  df-acs 16170
This theorem is referenced by:  acsfiel2  16237  isacs3lem  17087
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