Theorem fgfil 23899

Description: A filter generates itself. (Contributed by Jeff Hankins, 5-Sep-2009.) (Revised by Stefan O'Rear, 2-Aug-2015.)

Assertion

Ref	Expression
fgfil	⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)

Proof of Theorem fgfil

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

Step	Hyp	Ref	Expression
1		filfbas 23872	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2		elfg 23895	. . . . 5 ⊢ (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)))
3	1, 2	syl 17	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡)))
4		filss 23877	. . . . . . . 8 ⊢ ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥 ∈ 𝐹 ∧ 𝑡 ⊆ 𝑋 ∧ 𝑥 ⊆ 𝑡)) → 𝑡 ∈ 𝐹)
5	4	3exp2 1353	. . . . . . 7 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → (𝑡 ⊆ 𝑋 → (𝑥 ⊆ 𝑡 → 𝑡 ∈ 𝐹))))
6	5	com34 91	. . . . . 6 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑥 ∈ 𝐹 → (𝑥 ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐹))))
7	6	rexlimdv 3151	. . . . 5 ⊢ (𝐹 ∈ (Fil‘𝑋) → (∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡 → (𝑡 ⊆ 𝑋 → 𝑡 ∈ 𝐹)))
8	7	impcomd 411	. . . 4 ⊢ (𝐹 ∈ (Fil‘𝑋) → ((𝑡 ⊆ 𝑋 ∧ ∃𝑥 ∈ 𝐹 𝑥 ⊆ 𝑡) → 𝑡 ∈ 𝐹))
9	3, 8	sylbid 240	. . 3 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) → 𝑡 ∈ 𝐹))
10	9	ssrdv 4001	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) ⊆ 𝐹)
11		ssfg 23896	. . 3 ⊢ (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
12	1, 11	syl 17	. 2 ⊢ (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
13	10, 12	eqssd 4013	1 ⊢ (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ 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:  elfilss  23900  fgtr  23914  fmid  23984  isfcf  24058  cnextcn  24091  filnetlem4  36364