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Theorem acsfiel 17539
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 17537 . . . . 5 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
2 mress 17478 . . . . 5 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) β†’ 𝑆 βŠ† 𝑋)
31, 2sylan 581 . . . 4 ((𝐢 ∈ (ACSβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) β†’ 𝑆 βŠ† 𝑋)
43ex 414 . . 3 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝑆 ∈ 𝐢 β†’ 𝑆 βŠ† 𝑋))
54pm4.71rd 564 . 2 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝑆 ∈ 𝐢 ↔ (𝑆 βŠ† 𝑋 ∧ 𝑆 ∈ 𝐢)))
6 eleq1 2822 . . . . 5 (𝑠 = 𝑆 β†’ (𝑠 ∈ 𝐢 ↔ 𝑆 ∈ 𝐢))
7 pweq 4575 . . . . . . 7 (𝑠 = 𝑆 β†’ 𝒫 𝑠 = 𝒫 𝑆)
87ineq1d 4172 . . . . . 6 (𝑠 = 𝑆 β†’ (𝒫 𝑠 ∩ Fin) = (𝒫 𝑆 ∩ Fin))
9 sseq2 3971 . . . . . 6 (𝑠 = 𝑆 β†’ ((πΉβ€˜π‘¦) βŠ† 𝑠 ↔ (πΉβ€˜π‘¦) βŠ† 𝑆))
108, 9raleqbidv 3318 . . . . 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 346 . . . 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 17538 . . . . . 6 (𝐢 ∈ (ACSβ€˜π‘‹) ↔ (𝐢 ∈ (Mooreβ€˜π‘‹) ∧ βˆ€π‘  ∈ 𝒫 𝑋(𝑠 ∈ 𝐢 ↔ βˆ€π‘¦ ∈ (𝒫 𝑠 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑠)))
1413simprbi 498 . . . . 5 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ βˆ€π‘  ∈ 𝒫 𝑋(𝑠 ∈ 𝐢 ↔ βˆ€π‘¦ ∈ (𝒫 𝑠 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑠))
1514adantr 482 . . . 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 6880 . . . . . 6 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ 𝑋 ∈ dom ACS)
17 elpw2g 5302 . . . . . 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 479 . . . 4 ((𝐢 ∈ (ACSβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ 𝑆 ∈ 𝒫 𝑋)
2011, 15, 19rspcdva 3581 . . 3 ((𝐢 ∈ (ACSβ€˜π‘‹) ∧ 𝑆 βŠ† 𝑋) β†’ (𝑆 ∈ 𝐢 ↔ βˆ€π‘¦ ∈ (𝒫 𝑆 ∩ Fin)(πΉβ€˜π‘¦) βŠ† 𝑆))
2120pm5.32da 580 . 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  dom cdm 5634  β€˜cfv 6497  Fincfn 8886  Moorecmre 17467  mrClscmrc 17468  ACScacs 17470
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-mre 17471  df-mrc 17472  df-acs 17474
This theorem is referenced by:  acsfiel2  17540  isacs3lem  18436
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