Theorem fgmin 36353

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 23895	. . . . . . 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 4065	. . . . . . . . 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 23877	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑡 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑡)) → 𝑡 ∈ 𝐹)
7	6	3exp2 1353	. . . . . . . . . . 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 3151	. . . . . . . . 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 4001	. . 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 23896	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵))
18		sstr2 4002	. . . 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 2106  ∃wrex 3068   ⊆ wss 3963  ‘cfv 6563  (class class class)co 7431  fBascfbas 21370  filGencfg 21371  Filcfil 23869

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-fbas 21379  df-fg 21380  df-fil 23870

This theorem is referenced by: (None)