Theorem fgmin 36371

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)) → (𝐵 ⊆ 𝐹 ↔ (𝑋filGen𝐵) ⊆ 𝐹))

Proof of Theorem fgmin

Dummy variables 𝑥 𝑡 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		elfg 23879	. . . . . . 7 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡)))
2	1	adantr 480	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡)))
3	2	adantr 480	. . . . 5 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵 ⊆ 𝐹) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡)))
4		ssrexv 4053	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐹 → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))
5	4	adantl 481	. . . . . . . 8 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵 ⊆ 𝐹) → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))
6		filss 23861	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑡 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑡)) → 𝑡 ∈ 𝐹)
7	6	3exp2 1355	. . . . . . . . . . 11 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → (𝑡 ⊆ 𝑋 → (𝑥 ⊆ 𝑡 → 𝑡 ∈ 𝐹))))
8	7	com34 91	. . . . . . . . . 10 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → (𝑥 ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐹))))
9	8	rexlimdv 3153	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐹)))
10	9	ad2antlr 727	. . . . . . . 8 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵 ⊆ 𝐹) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐹)))
11	5, 10	syld 47	. . . . . . 7 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵 ⊆ 𝐹) → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐹)))
12	11	com23 86	. . . . . 6 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵 ⊆ 𝐹) → (𝑡 ⊆ 𝑋 → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡 → 𝑡 ∈ 𝐹)))
13	12	impd 410	. . . . 5 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵 ⊆ 𝐹) → ((𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡) → 𝑡 ∈ 𝐹))
14	3, 13	sylbid 240	. . . 4 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵 ⊆ 𝐹) → (𝑡 ∈ (𝑋filGen𝐵) → 𝑡 ∈ 𝐹))
15	14	ssrdv 3989	. . 3 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵 ⊆ 𝐹) → (𝑋filGen𝐵) ⊆ 𝐹)
16	15	ex 412	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵 ⊆ 𝐹 → (𝑋filGen𝐵) ⊆ 𝐹))
17		ssfg 23880	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵))
18		sstr2 3990	. . . 4 ⊢ (𝐵 ⊆ (𝑋filGen𝐵) → ((𝑋filGen𝐵) ⊆ 𝐹 → 𝐵 ⊆ 𝐹))
19	17, 18	syl 17	. . 3 ⊢ (𝐵 ∈ (fBas‘𝑋) → ((𝑋filGen𝐵) ⊆ 𝐹 → 𝐵 ⊆ 𝐹))
20	19	adantr 480	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝑋filGen𝐵) ⊆ 𝐹 → 𝐵 ⊆ 𝐹))
21	16, 20	impbid 212	1 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵 ⊆ 𝐹 ↔ (𝑋filGen𝐵) ⊆ 𝐹))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∈ wcel 2108  ∃wrex 3070   ⊆ wss 3951  ‘cfv 6561  (class class class)co 7431  fBascfbas 21352  filGencfg 21353  Filcfil 23853

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-pw 4602  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fv 6569  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21361  df-fg 21362  df-fil 23854

This theorem is referenced by: (None)