Theorem fgmin 36365

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 23765	. . . . . . 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 4019	. . . . . . . . 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 23747	. . . . . . . . . . . 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 3133	. . . . . . . . 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 3955	. . 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 23766	. . . 4 ⊢ (𝐵 ∈ (fBas‘𝑋) → 𝐵 ⊆ (𝑋filGen𝐵))
18		sstr2 3956	. . . 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 2109  ∃wrex 3054   ⊆ wss 3917  ‘cfv 6514  (class class class)co 7390  fBascfbas 21259  filGencfg 21260  Filcfil 23739

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fv 6522  df-ov 7393  df-oprab 7394  df-mpo 7395  df-fbas 21268  df-fg 21269  df-fil 23740

This theorem is referenced by: (None)