MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  acsfiel Structured version   Visualization version   GIF version

Theorem acsfiel 16515
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 16513 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → 𝐶 ∈ (Moore‘𝑋))
2 mress 16454 . . . . 5 ((𝐶 ∈ (Moore‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
31, 2sylan 569 . . . 4 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝐶) → 𝑆𝑋)
43ex 397 . . 3 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶𝑆𝑋))
54pm4.71rd 552 . 2 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶 ↔ (𝑆𝑋𝑆𝐶)))
6 eleq1 2838 . . . . 5 (𝑠 = 𝑆 → (𝑠𝐶𝑆𝐶))
7 pweq 4300 . . . . . . 7 (𝑠 = 𝑆 → 𝒫 𝑠 = 𝒫 𝑆)
87ineq1d 3964 . . . . . 6 (𝑠 = 𝑆 → (𝒫 𝑠 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
9 sseq2 3776 . . . . . 6 (𝑠 = 𝑆 → ((𝐹𝑦) ⊆ 𝑠 ↔ (𝐹𝑦) ⊆ 𝑆))
108, 9raleqbidv 3301 . . . . 5 (𝑠 = 𝑆 → (∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))
116, 10bibi12d 334 . . . 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 16514 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) ↔ (𝐶 ∈ (Moore‘𝑋) ∧ ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠)))
1413simprbi 484 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))
1514adantr 466 . . . 4 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → ∀𝑠 ∈ 𝒫 𝑋(𝑠𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑠 ∩ Fin)(𝐹𝑦) ⊆ 𝑠))
16 elfvdm 6359 . . . . . 6 (𝐶 ∈ (ACS‘𝑋) → 𝑋 ∈ dom ACS)
17 elpw2g 4958 . . . . . 6 (𝑋 ∈ dom ACS → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1816, 17syl 17 . . . . 5 (𝐶 ∈ (ACS‘𝑋) → (𝑆 ∈ 𝒫 𝑋𝑆𝑋))
1918biimpar 463 . . . 4 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → 𝑆 ∈ 𝒫 𝑋)
2011, 15, 19rspcdva 3466 . . 3 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))
2120pm5.32da 568 . 2 (𝐶 ∈ (ACS‘𝑋) → ((𝑆𝑋𝑆𝐶) ↔ (𝑆𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆)))
225, 21bitrd 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 382   = wceq 1631  wcel 2145  wral 3061  cin 3722  wss 3723  𝒫 cpw 4297  dom cdm 5249  cfv 6029  Fincfn 8107  Moorecmre 16443  mrClscmrc 16444  ACScacs 16446
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7094
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-int 4612  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5992  df-fun 6031  df-fn 6032  df-f 6033  df-fv 6037  df-mre 16447  df-mrc 16448  df-acs 16450
This theorem is referenced by:  acsfiel2  16516  isacs3lem  17367
  Copyright terms: Public domain W3C validator