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Theorem acsfiel 17604
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 = (mrClsβ€˜πΆ)
Assertion
Ref Expression
acsfiel (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝑆 ∈ 𝐢 ↔ (𝑆 βŠ† 𝑋 ∧ βˆ€π‘¦ ∈ (𝒫 𝑆 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑆)))
Distinct variable groups:   𝑦,𝐢   𝑦,𝐹   𝑦,𝑆   𝑦,𝑋

Proof of Theorem acsfiel
Dummy variable 𝑠 is distinct from all other variables.
StepHypRef Expression
1 acsmre 17602 . . . . 5 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
2 mress 17543 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) β†’ 𝑆 βŠ† 𝑋)
31, 2sylan 579 . . . 4 ((𝐢 ∈ (ACSβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) β†’ 𝑆 βŠ† 𝑋)
43ex 412 . . 3 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝑆 ∈ 𝐢 β†’ 𝑆 βŠ† 𝑋))
54pm4.71rd 562 . 2 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝑆 ∈ 𝐢 ↔ (𝑆 βŠ† 𝑋 ∧ 𝑆 ∈ 𝐢)))
6 eleq1 2815 . . . . 5 (𝑠 = 𝑆 β†’ (𝑠 ∈ 𝐢 ↔ 𝑆 ∈ 𝐢))
7 pweq 4611 . . . . . . 7 (𝑠 = 𝑆 β†’ 𝒫 𝑠 = 𝒫 𝑆)
87ineq1d 4206 . . . . . 6 (𝑠 = 𝑆 β†’ (𝒫 𝑠 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
9 sseq2 4003 . . . . . 6 (𝑠 = 𝑆 β†’ ((πΉβ€˜π‘¦) βŠ† 𝑠 ↔ (πΉβ€˜π‘¦) βŠ† 𝑆))
108, 9raleqbidv 3336 . . . . 5 (𝑠 = 𝑆 β†’ (βˆ€π‘¦ ∈ (𝒫 𝑠 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑠 ↔ βˆ€π‘¦ ∈ (𝒫 𝑆 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑆))
116, 10bibi12d 345 . . . 4 (𝑠 = 𝑆 β†’ ((𝑠 ∈ 𝐢 ↔ βˆ€π‘¦ ∈ (𝒫 𝑠 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑠) ↔ (𝑆 ∈ 𝐢 ↔ βˆ€π‘¦ ∈ (𝒫 𝑆 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑆)))
12 isacs2.f . . . . . . 7 𝐹 = (mrClsβ€˜πΆ)
1312isacs2 17603 . . . . . 6 (𝐢 ∈ (ACSβ€˜π‘‹) ↔ (𝐢 ∈ (Mooreβ€˜π‘‹) ∧ βˆ€π‘  ∈ 𝒫 𝑋(𝑠 ∈ 𝐢 ↔ βˆ€π‘¦ ∈ (𝒫 𝑠 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑠)))
1413simprbi 496 . . . . 5 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ βˆ€π‘  ∈ 𝒫 𝑋(𝑠 ∈ 𝐢 ↔ βˆ€π‘¦ ∈ (𝒫 𝑠 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑠))
1514adantr 480 . . . 4 ((𝐢 ∈ (ACSβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ βˆ€π‘  ∈ 𝒫 𝑋(𝑠 ∈ 𝐢 ↔ βˆ€π‘¦ ∈ (𝒫 𝑠 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑠))
16 elfvdm 6921 . . . . . 6 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ 𝑋 ∈ dom ACS)
17 elpw2g 5337 . . . . . 6 (𝑋 ∈ dom ACS β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
1816, 17syl 17 . . . . 5 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝑆 ∈ 𝒫 𝑋 ↔ 𝑆 βŠ† 𝑋))
1918biimpar 477 . . . 4 ((𝐢 ∈ (ACSβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 ∈ 𝒫 𝑋)
2011, 15, 19rspcdva 3607 . . 3 ((𝐢 ∈ (ACSβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 ∈ 𝐢 ↔ βˆ€π‘¦ ∈ (𝒫 𝑆 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑆))
2120pm5.32da 578 . 2 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ ((𝑆 βŠ† 𝑋 ∧ 𝑆 ∈ 𝐢) ↔ (𝑆 βŠ† 𝑋 ∧ βˆ€π‘¦ ∈ (𝒫 𝑆 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑆)))
225, 21bitrd 279 1 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝑆 ∈ 𝐢 ↔ (𝑆 βŠ† 𝑋 ∧ βˆ€π‘¦ ∈ (𝒫 𝑆 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑆)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  βˆ€wral 3055   ∩ cin 3942   βŠ† wss 3943  π’« cpw 4597  dom cdm 5669  β€˜cfv 6536  Fincfn 8938  Moorecmre 17532  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:  acsfiel2  17605  isacs3lem  18504
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