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Theorem mrefg3 42005
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 42004 . . 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 3992 . . . 4 (𝑆 = (πΉβ€˜π‘”) ↔ (𝑆 βŠ† (πΉβ€˜π‘”) ∧ (πΉβ€˜π‘”) βŠ† 𝑆))
5 simpll 764 . . . . . 6 (((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) β†’ 𝐢 ∈ (Mooreβ€˜π‘‹))
6 inss1 4223 . . . . . . . . 9 (𝒫 𝑆 ∩ Fin) βŠ† 𝒫 𝑆
76sseli 3973 . . . . . . . 8 (𝑔 ∈ (𝒫 𝑆 ∩ Fin) β†’ 𝑔 ∈ 𝒫 𝑆)
87elpwid 4606 . . . . . . 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 766 . . . . . 6 (((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑆 ∈ 𝐢) ∧ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) β†’ 𝑆 ∈ 𝐢)
111mrcsscl 17571 . . . . . 6 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑆 ∧ 𝑆 ∈ 𝐢) β†’ (πΉβ€˜π‘”) βŠ† 𝑆)
125, 9, 10, 11syl3anc 1368 . . . . 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 3170 . 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 1533   ∈ wcel 2098  βˆƒwrex 3064   ∩ cin 3942   βŠ† wss 3943  π’« cpw 4597  β€˜cfv 6536  Fincfn 8938  Moorecmre 17533  mrClscmrc 17534
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544  df-mre 17537  df-mrc 17538
This theorem is referenced by: (None)
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