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Theorem fgmin 33803
Description: Minimality property of a generated filter: every filter that contains 𝐵 contains its generated filter. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Mario Carneiro, 7-Aug-2015.)
Assertion
Ref Expression
fgmin ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵𝐹 ↔ (𝑋filGen𝐵) ⊆ 𝐹))

Proof of Theorem fgmin
Dummy variables 𝑥 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfg 22485 . . . . . . 7 (𝐵 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡)))
21adantr 484 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡)))
32adantr 484 . . . . 5 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡)))
4 ssrexv 4020 . . . . . . . . 9 (𝐵𝐹 → (∃𝑥𝐵 𝑥𝑡 → ∃𝑥𝐹 𝑥𝑡))
54adantl 485 . . . . . . . 8 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (∃𝑥𝐵 𝑥𝑡 → ∃𝑥𝐹 𝑥𝑡))
6 filss 22467 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹𝑡𝑋𝑥𝑡)) → 𝑡𝐹)
763exp2 1351 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑡𝑋 → (𝑥𝑡𝑡𝐹))))
87com34 91 . . . . . . . . . 10 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑥𝑡 → (𝑡𝑋𝑡𝐹))))
98rexlimdv 3275 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → (∃𝑥𝐹 𝑥𝑡 → (𝑡𝑋𝑡𝐹)))
109ad2antlr 726 . . . . . . . 8 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (∃𝑥𝐹 𝑥𝑡 → (𝑡𝑋𝑡𝐹)))
115, 10syld 47 . . . . . . 7 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (∃𝑥𝐵 𝑥𝑡 → (𝑡𝑋𝑡𝐹)))
1211com23 86 . . . . . 6 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (𝑡𝑋 → (∃𝑥𝐵 𝑥𝑡𝑡𝐹)))
1312impd 414 . . . . 5 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → ((𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡) → 𝑡𝐹))
143, 13sylbid 243 . . . 4 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (𝑡 ∈ (𝑋filGen𝐵) → 𝑡𝐹))
1514ssrdv 3959 . . 3 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (𝑋filGen𝐵) ⊆ 𝐹)
1615ex 416 . 2 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵𝐹 → (𝑋filGen𝐵) ⊆ 𝐹))
17 ssfg 22486 . . . 4 (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵))
18 sstr2 3960 . . . 4 (𝐵 ⊆ (𝑋filGen𝐵) → ((𝑋filGen𝐵) ⊆ 𝐹𝐵𝐹))
1917, 18syl 17 . . 3 (𝐵 ∈ (fBas‘𝑋) → ((𝑋filGen𝐵) ⊆ 𝐹𝐵𝐹))
2019adantr 484 . 2 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝑋filGen𝐵) ⊆ 𝐹𝐵𝐹))
2116, 20impbid 215 1 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵𝐹 ↔ (𝑋filGen𝐵) ⊆ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wcel 2115  wrex 3134  wss 3919  cfv 6345  (class class class)co 7151  fBascfbas 20088  filGencfg 20089  Filcfil 22459
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-nel 3119  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-op 4557  df-uni 4825  df-br 5054  df-opab 5116  df-mpt 5134  df-id 5448  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-iota 6304  df-fun 6347  df-fv 6353  df-ov 7154  df-oprab 7155  df-mpo 7156  df-fbas 20097  df-fg 20098  df-fil 22460
This theorem is referenced by: (None)
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