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Theorem fgmin 32490
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 21722 . . . . . . 7 (𝐵 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡)))
21adantr 480 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡)))
32adantr 480 . . . . 5 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡)))
4 ssrexv 3700 . . . . . . . . 9 (𝐵𝐹 → (∃𝑥𝐵 𝑥𝑡 → ∃𝑥𝐹 𝑥𝑡))
54adantl 481 . . . . . . . 8 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (∃𝑥𝐵 𝑥𝑡 → ∃𝑥𝐹 𝑥𝑡))
6 filss 21704 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹𝑡𝑋𝑥𝑡)) → 𝑡𝐹)
763exp2 1307 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑡𝑋 → (𝑥𝑡𝑡𝐹))))
87com34 91 . . . . . . . . . 10 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑥𝑡 → (𝑡𝑋𝑡𝐹))))
98rexlimdv 3059 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → (∃𝑥𝐹 𝑥𝑡 → (𝑡𝑋𝑡𝐹)))
109ad2antlr 763 . . . . . . . 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 446 . . . . 5 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → ((𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡) → 𝑡𝐹))
143, 13sylbid 230 . . . 4 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (𝑡 ∈ (𝑋filGen𝐵) → 𝑡𝐹))
1514ssrdv 3642 . . 3 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (𝑋filGen𝐵) ⊆ 𝐹)
1615ex 449 . 2 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵𝐹 → (𝑋filGen𝐵) ⊆ 𝐹))
17 ssfg 21723 . . . 4 (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵))
18 sstr2 3643 . . . 4 (𝐵 ⊆ (𝑋filGen𝐵) → ((𝑋filGen𝐵) ⊆ 𝐹𝐵𝐹))
1917, 18syl 17 . . 3 (𝐵 ∈ (fBas‘𝑋) → ((𝑋filGen𝐵) ⊆ 𝐹𝐵𝐹))
2019adantr 480 . 2 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝑋filGen𝐵) ⊆ 𝐹𝐵𝐹))
2116, 20impbid 202 1 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵𝐹 ↔ (𝑋filGen𝐵) ⊆ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2030  wrex 2942  wss 3607  cfv 5926  (class class class)co 6690  fBascfbas 19782  filGencfg 19783  Filcfil 21696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-nel 2927  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-fbas 19791  df-fg 19792  df-fil 21697
This theorem is referenced by: (None)
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