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Theorem isnacs2 41434
Description: Express Noetherian-type closure system with fewer quantifiers. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypothesis
Ref Expression
isnacs.f 𝐹 = (mrClsβ€˜πΆ)
Assertion
Ref Expression
isnacs2 (𝐢 ∈ (NoeACSβ€˜π‘‹) ↔ (𝐢 ∈ (ACSβ€˜π‘‹) ∧ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) = 𝐢))

Proof of Theorem isnacs2
Dummy variables 𝑔 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isnacs.f . . 3 𝐹 = (mrClsβ€˜πΆ)
21isnacs 41432 . 2 (𝐢 ∈ (NoeACSβ€˜π‘‹) ↔ (𝐢 ∈ (ACSβ€˜π‘‹) ∧ βˆ€π‘  ∈ 𝐢 βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (πΉβ€˜π‘”)))
3 eqcom 2739 . . . . . . . 8 (𝑠 = (πΉβ€˜π‘”) ↔ (πΉβ€˜π‘”) = 𝑠)
43rexbii 3094 . . . . . . 7 (βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (πΉβ€˜π‘”) ↔ βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)(πΉβ€˜π‘”) = 𝑠)
5 acsmre 17595 . . . . . . . . 9 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
61mrcf 17552 . . . . . . . . 9 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐹:𝒫 π‘‹βŸΆπΆ)
7 ffn 6717 . . . . . . . . 9 (𝐹:𝒫 π‘‹βŸΆπΆ β†’ 𝐹 Fn 𝒫 𝑋)
85, 6, 73syl 18 . . . . . . . 8 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ 𝐹 Fn 𝒫 𝑋)
9 inss1 4228 . . . . . . . 8 (𝒫 𝑋 ∩ Fin) βŠ† 𝒫 𝑋
10 fvelimab 6964 . . . . . . . 8 ((𝐹 Fn 𝒫 𝑋 ∧ (𝒫 𝑋 ∩ Fin) βŠ† 𝒫 𝑋) β†’ (𝑠 ∈ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) ↔ βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)(πΉβ€˜π‘”) = 𝑠))
118, 9, 10sylancl 586 . . . . . . 7 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝑠 ∈ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) ↔ βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)(πΉβ€˜π‘”) = 𝑠))
124, 11bitr4id 289 . . . . . 6 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (πΉβ€˜π‘”) ↔ 𝑠 ∈ (𝐹 β€œ (𝒫 𝑋 ∩ Fin))))
1312ralbidv 3177 . . . . 5 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (βˆ€π‘  ∈ 𝐢 βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (πΉβ€˜π‘”) ↔ βˆ€π‘  ∈ 𝐢 𝑠 ∈ (𝐹 β€œ (𝒫 𝑋 ∩ Fin))))
14 dfss3 3970 . . . . 5 (𝐢 βŠ† (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) ↔ βˆ€π‘  ∈ 𝐢 𝑠 ∈ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)))
1513, 14bitr4di 288 . . . 4 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (βˆ€π‘  ∈ 𝐢 βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (πΉβ€˜π‘”) ↔ 𝐢 βŠ† (𝐹 β€œ (𝒫 𝑋 ∩ Fin))))
16 imassrn 6070 . . . . . . 7 (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) βŠ† ran 𝐹
17 frn 6724 . . . . . . . 8 (𝐹:𝒫 π‘‹βŸΆπΆ β†’ ran 𝐹 βŠ† 𝐢)
185, 6, 173syl 18 . . . . . . 7 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ ran 𝐹 βŠ† 𝐢)
1916, 18sstrid 3993 . . . . . 6 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) βŠ† 𝐢)
2019biantrurd 533 . . . . 5 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝐢 βŠ† (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) ↔ ((𝐹 β€œ (𝒫 𝑋 ∩ Fin)) βŠ† 𝐢 ∧ 𝐢 βŠ† (𝐹 β€œ (𝒫 𝑋 ∩ Fin)))))
21 eqss 3997 . . . . 5 ((𝐹 β€œ (𝒫 𝑋 ∩ Fin)) = 𝐢 ↔ ((𝐹 β€œ (𝒫 𝑋 ∩ Fin)) βŠ† 𝐢 ∧ 𝐢 βŠ† (𝐹 β€œ (𝒫 𝑋 ∩ Fin))))
2220, 21bitr4di 288 . . . 4 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝐢 βŠ† (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) ↔ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) = 𝐢))
2315, 22bitrd 278 . . 3 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (βˆ€π‘  ∈ 𝐢 βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (πΉβ€˜π‘”) ↔ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) = 𝐢))
2423pm5.32i 575 . 2 ((𝐢 ∈ (ACSβ€˜π‘‹) ∧ βˆ€π‘  ∈ 𝐢 βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (πΉβ€˜π‘”)) ↔ (𝐢 ∈ (ACSβ€˜π‘‹) ∧ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) = 𝐢))
252, 24bitri 274 1 (𝐢 ∈ (NoeACSβ€˜π‘‹) ↔ (𝐢 ∈ (ACSβ€˜π‘‹) ∧ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) = 𝐢))
Colors of variables: wff setvar class
Syntax hints:   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3947   βŠ† wss 3948  π’« cpw 4602  ran crn 5677   β€œ cima 5679   Fn wfn 6538  βŸΆwf 6539  β€˜cfv 6543  Fincfn 8938  Moorecmre 17525  mrClscmrc 17526  ACScacs 17528  NoeACScnacs 41430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fv 6551  df-mre 17529  df-mrc 17530  df-acs 17532  df-nacs 41431
This theorem is referenced by:  nacsacs  41437
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