Theorem fgmin 33268

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 22195	. . . . . . 7 ⊢ (𝐵 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡)))
2	1	adantr 473	. . . . . 6 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡)))
3	2	adantr 473	. . . . 5 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵 ⊆ 𝐹) → (𝑡 ∈ (𝑋filGen𝐵) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡)))
4		ssrexv 3918	. . . . . . . . 9 ⊢ (𝐵 ⊆ 𝐹 → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))
5	4	adantl 474	. . . . . . . 8 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵 ⊆ 𝐹) → (∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡 → ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡))
6		filss 22177	. . . . . . . . . . . 12 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑡 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑡)) → 𝑡 ∈ 𝐹)
7	6	3exp2 1334	. . . . . . . . . . 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 3222	. . . . . . . . 9 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐹)))
10	9	ad2antlr 714	. . . . . . . 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 402	. . . . 5 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵 ⊆ 𝐹) → ((𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐵 𝑥 ⊆ 𝑡) → 𝑡 ∈ 𝐹))
14	3, 13	sylbid 232	. . . 4 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵 ⊆ 𝐹) → (𝑡 ∈ (𝑋filGen𝐵) → 𝑡 ∈ 𝐹))
15	14	ssrdv 3858	. . 3 ⊢ (((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) ∧ 𝐵 ⊆ 𝐹) → (𝑋filGen𝐵) ⊆ 𝐹)
16	15	ex 405	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵 ⊆ 𝐹 → (𝑋filGen𝐵) ⊆ 𝐹))
17		ssfg 22196	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵))
18		sstr2 3859	. . . 4 ⊢ (𝐵 ⊆ (𝑋filGen𝐵) → ((𝑋filGen𝐵) ⊆ 𝐹 → 𝐵 ⊆ 𝐹))
19	17, 18	syl 17	. . 3 ⊢ (𝐵 ∈ (fBas‘𝑋) → ((𝑋filGen𝐵) ⊆ 𝐹 → 𝐵 ⊆ 𝐹))
20	19	adantr 473	. 2 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → ((𝑋filGen𝐵) ⊆ 𝐹 → 𝐵 ⊆ 𝐹))
21	16, 20	impbid 204	1 ⊢ ((𝐵 ∈ (fBas‘𝑋) ∧ 𝐹 ∈ (Fil‘𝑋)) → (𝐵 ⊆ 𝐹 ↔ (𝑋filGen𝐵) ⊆ 𝐹))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   ∈ wcel 2050  ∃wrex 3083   ⊆ wss 3823  ‘cfv 6185  (class class class)co 6974  fBascfbas 20247  filGencfg 20248  Filcfil 22169

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2744  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2753  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-nel 3068  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3676  df-csb 3781  df-dif 3826  df-un 3828  df-in 3830  df-ss 3837  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4709  df-br 4926  df-opab 4988  df-mpt 5005  df-id 5308  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-iota 6149  df-fun 6187  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-fbas 20256  df-fg 20257  df-fil 22170

This theorem is referenced by: (None)