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Theorem acsfiel2 17605
Description: A set is closed in an algebraic closure system iff it contains all closures of finite subsets. (Contributed by Stefan O'Rear, 3-Apr-2015.)
Hypothesis
Ref Expression
isacs2.f 𝐹 = (mrCls‘𝐶)
Assertion
Ref Expression
acsfiel2 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))
Distinct variable groups:   𝑦,𝐶   𝑦,𝐹   𝑦,𝑆   𝑦,𝑋

Proof of Theorem acsfiel2
StepHypRef Expression
1 isacs2.f . . 3 𝐹 = (mrCls‘𝐶)
21acsfiel 17604 . 2 (𝐶 ∈ (ACS‘𝑋) → (𝑆𝐶 ↔ (𝑆𝑋 ∧ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆)))
32baibd 539 1 ((𝐶 ∈ (ACS‘𝑋) ∧ 𝑆𝑋) → (𝑆𝐶 ↔ ∀𝑦 ∈ (𝒫 𝑆 ∩ Fin)(𝐹𝑦) ⊆ 𝑆))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  wral 3055  cin 3942  wss 3943  𝒫 cpw 4597  cfv 6536  Fincfn 8938  mrClscmrc 17533  ACScacs 17535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-mre 17536  df-mrc 17537  df-acs 17539
This theorem is referenced by:  mreacs  17608
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