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Theorem fgmin 31399
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 21409 . . . . . . 7 (𝐵 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡)))
21adantr 479 . . . . . 6 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡)))
32adantr 479 . . . . 5 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡)))
4 ssrexv 3534 . . . . . . . . 9 (𝐵𝐹 → (∃𝑥𝐵 𝑥𝑡 → ∃𝑥𝐹 𝑥𝑡))
54adantl 480 . . . . . . . 8 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (∃𝑥𝐵 𝑥𝑡 → ∃𝑥𝐹 𝑥𝑡))
6 filss 21391 . . . . . . . . . . . 12 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹𝑡𝑋𝑥𝑡)) → 𝑡𝐹)
763exp2 1276 . . . . . . . . . . 11 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑡𝑋 → (𝑥𝑡𝑡𝐹))))
87com34 88 . . . . . . . . . 10 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑥𝑡 → (𝑡𝑋𝑡𝐹))))
98rexlimdv 2916 . . . . . . . . 9 (𝐹 ∈ (Fil‘𝑋) → (∃𝑥𝐹 𝑥𝑡 → (𝑡𝑋𝑡𝐹)))
109ad2antlr 758 . . . . . . . 8 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (∃𝑥𝐹 𝑥𝑡 → (𝑡𝑋𝑡𝐹)))
115, 10syld 45 . . . . . . 7 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (∃𝑥𝐵 𝑥𝑡 → (𝑡𝑋𝑡𝐹)))
1211com23 83 . . . . . 6 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (𝑡𝑋 → (∃𝑥𝐵 𝑥𝑡𝑡𝐹)))
1312impd 445 . . . . 5 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → ((𝑡𝑋 ∧ ∃𝑥𝐵 𝑥𝑡) → 𝑡𝐹))
143, 13sylbid 228 . . . 4 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (𝑡 ∈ (𝑋filGen𝐵) → 𝑡𝐹))
1514ssrdv 3478 . . 3 (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵𝐹) → (𝑋filGen𝐵) ⊆ 𝐹)
1615ex 448 . 2 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵𝐹 → (𝑋filGen𝐵) ⊆ 𝐹))
17 ssfg 21410 . . . 4 (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵))
18 sstr2 3479 . . . 4 (𝐵 ⊆ (𝑋filGen𝐵) → ((𝑋filGen𝐵) ⊆ 𝐹𝐵𝐹))
1917, 18syl 17 . . 3 (𝐵 ∈ (fBas‘𝑋) → ((𝑋filGen𝐵) ⊆ 𝐹𝐵𝐹))
2019adantr 479 . 2 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝑋filGen𝐵) ⊆ 𝐹𝐵𝐹))
2116, 20impbid 200 1 ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵𝐹 ↔ (𝑋filGen𝐵) ⊆ 𝐹))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wcel 1938  wrex 2801  wss 3444  cfv 5689  (class class class)co 6425  fBascfbas 19480  filGencfg 19481  Filcfil 21383
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pow 4668  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-nel 2687  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-csb 3404  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-mpt 4543  df-id 4847  df-xp 4938  df-rel 4939  df-cnv 4940  df-co 4941  df-dm 4942  df-rn 4943  df-res 4944  df-ima 4945  df-iota 5653  df-fun 5691  df-fv 5697  df-ov 6428  df-oprab 6429  df-mpt2 6430  df-fbas 19489  df-fg 19490  df-fil 21384
This theorem is referenced by: (None)
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