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Theorem mrefg3 41446
Description: Slight variation on finite generation for closure systems. (Contributed by Stefan O'Rear, 4-Apr-2015.)
Hypothesis
Ref Expression
isnacs.f 𝐹 = (mrClsβ€˜πΆ)
Assertion
Ref Expression
mrefg3 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) β†’ (βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (πΉβ€˜π‘”) ↔ βˆƒπ‘” ∈ (𝒫 𝑆 ∩ Fin)𝑆 βŠ† (πΉβ€˜π‘”)))
Distinct variable groups:   𝐢,𝑔   𝑔,𝐹   𝑆,𝑔   𝑔,𝑋
Proof of Theorem mrefg3
StepHypRef Expression
1 isnacs.f . . . 4 𝐹 = (mrClsβ€˜πΆ)
21mrefg2 41445 . . 3 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (πΉβ€˜π‘”) ↔ βˆƒπ‘” ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (πΉβ€˜π‘”)))
32adantr 482 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) β†’ (βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (πΉβ€˜π‘”) ↔ βˆƒπ‘” ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (πΉβ€˜π‘”)))
4 eqss 3998 . . . 4 (𝑆 = (πΉβ€˜π‘”) ↔ (𝑆 βŠ† (πΉβ€˜π‘”) ∧ (πΉβ€˜π‘”) βŠ† 𝑆))
5 simpll 766 . . . . . 6 (((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
6 inss1 4229 . . . . . . . . 9 (𝒫 𝑆 ∩ Fin) βŠ† 𝒫 𝑆
76sseli 3979 . . . . . . . 8 (𝑔 ∈ (𝒫 𝑆 ∩ Fin) β†’ 𝑔 ∈ 𝒫 𝑆)
87elpwid 4612 . . . . . . 7 (𝑔 ∈ (𝒫 𝑆 ∩ Fin) β†’ 𝑔 βŠ† 𝑆)
98adantl 483 . . . . . 6 (((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) β†’ 𝑔 βŠ† 𝑆)
10 simplr 768 . . . . . 6 (((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) β†’ 𝑆 ∈ 𝐢)
111mrcsscl 17564 . . . . . 6 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑆 ∧ 𝑆 ∈ 𝐢) β†’ (πΉβ€˜π‘”) βŠ† 𝑆)
125, 9, 10, 11syl3anc 1372 . . . . 5 (((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) β†’ (πΉβ€˜π‘”) βŠ† 𝑆)
1312biantrud 533 . . . 4 (((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) β†’ (𝑆 βŠ† (πΉβ€˜π‘”) ↔ (𝑆 βŠ† (πΉβ€˜π‘”) ∧ (πΉβ€˜π‘”) βŠ† 𝑆)))
144, 13bitr4id 290 . . 3 (((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) β†’ (𝑆 = (πΉβ€˜π‘”) ↔ 𝑆 βŠ† (πΉβ€˜π‘”)))
1514rexbidva 3177 . 2 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) β†’ (βˆƒπ‘” ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (πΉβ€˜π‘”) ↔ βˆƒπ‘” ∈ (𝒫 𝑆 ∩ Fin)𝑆 βŠ† (πΉβ€˜π‘”)))
163, 15bitrd 279 1 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) β†’ (βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (πΉβ€˜π‘”) ↔ βˆƒπ‘” ∈ (𝒫 𝑆 ∩ Fin)𝑆 βŠ† (πΉβ€˜π‘”)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3071   ∩ cin 3948   βŠ† wss 3949  π’« cpw 4603  β€˜cfv 6544  Fincfn 8939  Moorecmre 17526  mrClscmrc 17527
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-sbc 3779  df-csb 3895  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
This theorem is referenced by: (None)
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