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Theorem mrefg2 41430
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
mrefg2 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (πΉβ€˜π‘”) ↔ βˆƒπ‘” ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (πΉβ€˜π‘”)))
Distinct variable groups:   𝐢,𝑔   𝑔,𝐹   𝑆,𝑔   𝑔,𝑋
Proof of Theorem mrefg2
StepHypRef Expression
1 isnacs.f . . . . . . . . 9 𝐹 = (mrClsβ€˜πΆ)
21mrcssid 17557 . . . . . . . 8 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑋) β†’ 𝑔 βŠ† (πΉβ€˜π‘”))
3 simpr 485 . . . . . . . . 9 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑔 βŠ† (πΉβ€˜π‘”)) β†’ 𝑔 βŠ† (πΉβ€˜π‘”))
41mrcssv 17554 . . . . . . . . . 10 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (πΉβ€˜π‘”) βŠ† 𝑋)
54adantr 481 . . . . . . . . 9 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑔 βŠ† (πΉβ€˜π‘”)) β†’ (πΉβ€˜π‘”) βŠ† 𝑋)
63, 5sstrd 3991 . . . . . . . 8 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑔 βŠ† (πΉβ€˜π‘”)) β†’ 𝑔 βŠ† 𝑋)
72, 6impbida 799 . . . . . . 7 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (𝑔 βŠ† 𝑋 ↔ 𝑔 βŠ† (πΉβ€˜π‘”)))
8 vex 3478 . . . . . . . 8 𝑔 ∈ V
98elpw 4605 . . . . . . 7 (𝑔 ∈ 𝒫 𝑋 ↔ 𝑔 βŠ† 𝑋)
108elpw 4605 . . . . . . 7 (𝑔 ∈ 𝒫 (πΉβ€˜π‘”) ↔ 𝑔 βŠ† (πΉβ€˜π‘”))
117, 9, 103bitr4g 313 . . . . . 6 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (𝑔 ∈ 𝒫 𝑋 ↔ 𝑔 ∈ 𝒫 (πΉβ€˜π‘”)))
1211anbi1d 630 . . . . 5 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ ((𝑔 ∈ 𝒫 𝑋 ∧ 𝑔 ∈ Fin) ↔ (𝑔 ∈ 𝒫 (πΉβ€˜π‘”) ∧ 𝑔 ∈ Fin)))
13 elin 3963 . . . . 5 (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑔 ∈ 𝒫 𝑋 ∧ 𝑔 ∈ Fin))
14 elin 3963 . . . . 5 (𝑔 ∈ (𝒫 (πΉβ€˜π‘”) ∩ Fin) ↔ (𝑔 ∈ 𝒫 (πΉβ€˜π‘”) ∧ 𝑔 ∈ Fin))
1512, 13, 143bitr4g 313 . . . 4 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (πΉβ€˜π‘”) ∩ Fin)))
16 pweq 4615 . . . . . . 7 (𝑆 = (πΉβ€˜π‘”) β†’ 𝒫 𝑆 = 𝒫 (πΉβ€˜π‘”))
1716ineq1d 4210 . . . . . 6 (𝑆 = (πΉβ€˜π‘”) β†’ (𝒫 𝑆 ∩ Fin) = (𝒫 (πΉβ€˜π‘”) ∩ Fin))
1817eleq2d 2819 . . . . 5 (𝑆 = (πΉβ€˜π‘”) β†’ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (πΉβ€˜π‘”) ∩ Fin)))
1918bibi2d 342 . . . 4 (𝑆 = (πΉβ€˜π‘”) β†’ ((𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 𝑆 ∩ Fin)) ↔ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (πΉβ€˜π‘”) ∩ Fin))))
2015, 19syl5ibrcom 246 . . 3 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (𝑆 = (πΉβ€˜π‘”) β†’ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 𝑆 ∩ Fin))))
2120pm5.32rd 578 . 2 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ ((𝑔 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑆 = (πΉβ€˜π‘”)) ↔ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑆 = (πΉβ€˜π‘”))))
2221rexbidv2 3174 1 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (βˆƒπ‘” ∈ (𝒫 𝑋 ∩ Fin)𝑆 = (πΉβ€˜π‘”) ↔ βˆƒπ‘” ∈ (𝒫 𝑆 ∩ Fin)𝑆 = (πΉβ€˜π‘”)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106  βˆƒwrex 3070   ∩ cin 3946   βŠ† wss 3947  π’« cpw 4601  β€˜cfv 6540  Fincfn 8935  Moorecmre 17522  mrClscmrc 17523
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7721
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-mre 17526  df-mrc 17527
This theorem is referenced by:  mrefg3  41431  isnacs3  41433
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