Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  isnacs2 Structured version   Visualization version   GIF version

Theorem isnacs2 41492
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 41490 . 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 3095 . . . . . . 7 (βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑠 = (πΉβ€˜π‘”) ↔ βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)(πΉβ€˜π‘”) = 𝑠)
5 acsmre 17596 . . . . . . . . 9 (𝐢 ∈ (ACSβ€˜π‘‹) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
61mrcf 17553 . . . . . . . . 9 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ 𝐹:𝒫 π‘‹βŸΆπΆ)
7 ffn 6718 . . . . . . . . 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 4229 . . . . . . . 8 (𝒫 𝑋 ∩ Fin) βŠ† 𝒫 𝑋
10 fvelimab 6965 . . . . . . . 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 3178 . . . . 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 3971 . . . . 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 6071 . . . . . . 7 (𝐹 β€œ (𝒫 𝑋 ∩ Fin)) βŠ† ran 𝐹
17 frn 6725 . . . . . . . 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 3994 . . . . . 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 3998 . . . . 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 3062  βˆƒwrex 3071   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603  ran crn 5678   β€œ cima 5680   Fn wfn 6539  βŸΆwf 6540  β€˜cfv 6544  Fincfn 8939  Moorecmre 17526  mrClscmrc 17527  ACScacs 17529  NoeACScnacs 41488
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 5300  ax-nul 5307  ax-pow 5364  ax-pr 5428  ax-un 7725
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 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fv 6552  df-mre 17530  df-mrc 17531  df-acs 17533  df-nacs 41489
This theorem is referenced by:  nacsacs  41495
  Copyright terms: Public domain W3C validator