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Theorem fgmin 32340
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 21656 . . . . . . 7 (𝐵 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡)))
21adantr 481 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡)))
32adantr 481 . . . . 5 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡)))
4 ssrexv 3659 . . . . . . . . 9 (𝐵𝐹 → (∃𝑥𝐵 𝑥𝑡 → ∃𝑥𝐹 𝑥𝑡))
54adantl 482 . . . . . . . 8 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (∃𝑥𝐵 𝑥𝑡 → ∃𝑥𝐹 𝑥𝑡))
6 filss 21638 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹𝑡𝑋𝑥𝑡)) → 𝑡𝐹)
763exp2 1283 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑡𝑋 → (𝑥𝑡𝑡𝐹))))
87com34 91 . . . . . . . . . 10 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑥𝑡 → (𝑡𝑋𝑡𝐹))))
98rexlimdv 3026 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → (∃𝑥𝐹 𝑥𝑡 → (𝑡𝑋𝑡𝐹)))
109ad2antlr 762 . . . . . . . 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 447 . . . . 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 3601 . . 3 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (𝑋filGen𝐵) ⊆ 𝐹)
1615ex 450 . 2 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵𝐹 → (𝑋filGen𝐵) ⊆ 𝐹))
17 ssfg 21657 . . . 4 (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵))
18 sstr2 3602 . . . 4 (𝐵 ⊆ (𝑋filGen𝐵) → ((𝑋filGen𝐵) ⊆ 𝐹𝐵𝐹))
1917, 18syl 17 . . 3 (𝐵 ∈ (fBas‘𝑋) → ((𝑋filGen𝐵) ⊆ 𝐹𝐵𝐹))
2019adantr 481 . 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 384  wcel 1988  wrex 2910  wss 3567  cfv 5876  (class class class)co 6635  fBascfbas 19715  filGencfg 19716  Filcfil 21630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-nel 2895  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fv 5884  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-fbas 19724  df-fg 19725  df-fil 21631
This theorem is referenced by: (None)
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