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Theorem mrefg2 41073
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 17502 . . . . . . . 8 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑔 βŠ† 𝑋) β†’ 𝑔 βŠ† (πΉβ€˜π‘”))
3 simpr 486 . . . . . . . . 9 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑔 βŠ† (πΉβ€˜π‘”)) β†’ 𝑔 βŠ† (πΉβ€˜π‘”))
41mrcssv 17499 . . . . . . . . . 10 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (πΉβ€˜π‘”) βŠ† 𝑋)
54adantr 482 . . . . . . . . 9 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑔 βŠ† (πΉβ€˜π‘”)) β†’ (πΉβ€˜π‘”) βŠ† 𝑋)
63, 5sstrd 3955 . . . . . . . 8 ((𝐢 ∈ (Mooreβ€˜π‘‹) ∧ 𝑔 βŠ† (πΉβ€˜π‘”)) β†’ 𝑔 βŠ† 𝑋)
72, 6impbida 800 . . . . . . 7 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (𝑔 βŠ† 𝑋 ↔ 𝑔 βŠ† (πΉβ€˜π‘”)))
8 vex 3448 . . . . . . . 8 𝑔 ∈ V
98elpw 4565 . . . . . . 7 (𝑔 ∈ 𝒫 𝑋 ↔ 𝑔 βŠ† 𝑋)
108elpw 4565 . . . . . . 7 (𝑔 ∈ 𝒫 (πΉβ€˜π‘”) ↔ 𝑔 βŠ† (πΉβ€˜π‘”))
117, 9, 103bitr4g 314 . . . . . 6 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (𝑔 ∈ 𝒫 𝑋 ↔ 𝑔 ∈ 𝒫 (πΉβ€˜π‘”)))
1211anbi1d 631 . . . . 5 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ ((𝑔 ∈ 𝒫 𝑋 ∧ 𝑔 ∈ Fin) ↔ (𝑔 ∈ 𝒫 (πΉβ€˜π‘”) ∧ 𝑔 ∈ Fin)))
13 elin 3927 . . . . 5 (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ (𝑔 ∈ 𝒫 𝑋 ∧ 𝑔 ∈ Fin))
14 elin 3927 . . . . 5 (𝑔 ∈ (𝒫 (πΉβ€˜π‘”) ∩ Fin) ↔ (𝑔 ∈ 𝒫 (πΉβ€˜π‘”) ∧ 𝑔 ∈ Fin))
1512, 13, 143bitr4g 314 . . . 4 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (πΉβ€˜π‘”) ∩ Fin)))
16 pweq 4575 . . . . . . 7 (𝑆 = (πΉβ€˜π‘”) β†’ 𝒫 𝑆 = 𝒫 (πΉβ€˜π‘”))
1716ineq1d 4172 . . . . . 6 (𝑆 = (πΉβ€˜π‘”) β†’ (𝒫 𝑆 ∩ Fin) = (𝒫 (πΉβ€˜π‘”) ∩ Fin))
1817eleq2d 2820 . . . . 5 (𝑆 = (πΉβ€˜π‘”) β†’ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ↔ 𝑔 ∈ (𝒫 (πΉβ€˜π‘”) ∩ Fin)))
1918bibi2d 343 . . . 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 247 . . 3 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ (𝑆 = (πΉβ€˜π‘”) β†’ (𝑔 ∈ (𝒫 𝑋 ∩ Fin) ↔ 𝑔 ∈ (𝒫 𝑆 ∩ Fin))))
2120pm5.32rd 579 . 2 (𝐢 ∈ (Mooreβ€˜π‘‹) β†’ ((𝑔 ∈ (𝒫 𝑋 ∩ Fin) ∧ 𝑆 = (πΉβ€˜π‘”)) ↔ (𝑔 ∈ (𝒫 𝑆 ∩ Fin) ∧ 𝑆 = (πΉβ€˜π‘”))))
2221rexbidv2 3168 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 397   = wceq 1542   ∈ wcel 2107  βˆƒwrex 3070   ∩ cin 3910   βŠ† wss 3911  π’« cpw 4561  β€˜cfv 6497  Fincfn 8886  Moorecmre 17467  mrClscmrc 17468
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 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673
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 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-int 4909  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fv 6505  df-mre 17471  df-mrc 17472
This theorem is referenced by:  mrefg3  41074  isnacs3  41076
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