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Theorem mrefg3 42128
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 42127 . . 3 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (πΉβ€˜π‘”) ↔ βˆƒπ‘” ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (πΉβ€˜π‘”)))
32adantr 480 . 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 3995 . . . 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 3976 . . . . . . . 8 (𝑔 ∈ (𝒫 𝑆 ∩ Fin) β†’ 𝑔 ∈ 𝒫 𝑆)
87elpwid 4612 . . . . . . 7 (𝑔 ∈ (𝒫 𝑆 ∩ Fin) β†’ 𝑔 βŠ† 𝑆)
98adantl 481 . . . . . 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 17600 . . . . . 6 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑆 ∧ 𝑆 ∈ 𝐢) β†’ (πΉβ€˜π‘”) βŠ† 𝑆)
125, 9, 10, 11syl3anc 1369 . . . . 5 (((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) β†’ (πΉβ€˜π‘”) βŠ† 𝑆)
1312biantrud 531 . . . 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 3173 . 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 395   = wceq 1534   ∈ wcel 2099  βˆƒwrex 3067   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4603  β€˜cfv 6548  Fincfn 8964  Moorecmre 17562  mrClscmrc 17563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pow 5365  ax-pr 5429  ax-un 7740
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-int 4950  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-fv 6556  df-mre 17566  df-mrc 17567
This theorem is referenced by: (None)
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