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Theorem isnacs2 41076
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 41074 . 2 (𝐢 ∈ (NoeACSβ€˜π‘‹) ↔ (𝐢 ∈ (ACSβ€˜π‘‹) ∧ βˆ€π‘  ∈ 𝐢 βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (πΉβ€˜π‘”)))
3 eqcom 2740 . . . . . . . 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 17540 . . . . . . . . 9 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
61mrcf 17497 . . . . . . . . 9 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐹:𝒫 π‘‹βŸΆπΆ)
7 ffn 6672 . . . . . . . . 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 4192 . . . . . . . 8 (𝒫 𝑋 ∩ Fin) βŠ† 𝒫 𝑋
10 fvelimab 6918 . . . . . . . 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 587 . . . . . . 7 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝑠 ∈ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) ↔ βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)(πΉβ€˜π‘”) = 𝑠))
124, 11bitr4id 290 . . . . . 6 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (πΉβ€˜π‘”) ↔ 𝑠 ∈ (𝐹 β€œ (𝒫 𝑋 ∩ Fin))))
1312ralbidv 3171 . . . . 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 3936 . . . . 5 (𝐢 βŠ† (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) ↔ βˆ€π‘  ∈ 𝐢 𝑠 ∈ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)))
1513, 14bitr4di 289 . . . 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 6028 . . . . . . 7 (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) βŠ† ran 𝐹
17 frn 6679 . . . . . . . 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 3959 . . . . . 6 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) βŠ† 𝐢)
2019biantrurd 534 . . . . 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 3963 . . . . 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 289 . . . 4 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ (𝐢 βŠ† (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) ↔ (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) = 𝐢))
2315, 22bitrd 279 . . 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 576 . 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 275 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 397   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070   ∩ cin 3913   βŠ† wss 3914  π’« cpw 4564  ran crn 5638   β€œ cima 5640   Fn wfn 6495  βŸΆwf 6496  β€˜cfv 6500  Fincfn 8889  Moorecmre 17470  mrClscmrc 17471  ACScacs 17473  NoeACScnacs 41072
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 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676
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 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-br 5110  df-opab 5172  df-mpt 5193  df-id 5535  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-fv 6508  df-mre 17474  df-mrc 17475  df-acs 17477  df-nacs 41073
This theorem is referenced by:  nacsacs  41079
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