Theorem fgmin 36590

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 23830	. . . . . . 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 4005	. . . . . . . . 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 23812	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑡 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑡)) → 𝑡 ∈ 𝐹)
7	6	3exp2 1356	. . . . . . . . . . 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 3137	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐹)))
10	9	ad2antlr 728	. . . . . . . 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 3941	. . 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 23831	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵))
18		sstr2 3942	. . . 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 2114  ∃wrex 3062   ⊆ wss 3903  ‘cfv 6500  (class class class)co 7368  fBascfbas 21312  filGencfg 21313  Filcfil 23804

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pow 5312  ax-pr 5379

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-nel 3038  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-sbc 3743  df-csb 3852  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-iota 6456  df-fun 6502  df-fv 6508  df-ov 7371  df-oprab 7372  df-mpo 7373  df-fbas 21321  df-fg 21322  df-fil 23805

This theorem is referenced by: (None)