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Theorem fgfil 21619
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.
StepHypRef Expression
1 filfbas 21592 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ∈ (fBas‘𝑋))
2 elfg 21615 . . . . 5 (𝐹 ∈ (fBas‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
31, 2syl 17 . . . 4 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) ↔ (𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡)))
4 filss 21597 . . . . . . . . 9 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝑥𝐹𝑡𝑋𝑥𝑡)) → 𝑡𝐹)
543exp2 1282 . . . . . . . 8 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑡𝑋 → (𝑥𝑡𝑡𝐹))))
65com34 91 . . . . . . 7 (𝐹 ∈ (Fil‘𝑋) → (𝑥𝐹 → (𝑥𝑡 → (𝑡𝑋𝑡𝐹))))
76rexlimdv 3025 . . . . . 6 (𝐹 ∈ (Fil‘𝑋) → (∃𝑥𝐹 𝑥𝑡 → (𝑡𝑋𝑡𝐹)))
87com23 86 . . . . 5 (𝐹 ∈ (Fil‘𝑋) → (𝑡𝑋 → (∃𝑥𝐹 𝑥𝑡𝑡𝐹)))
98impd 447 . . . 4 (𝐹 ∈ (Fil‘𝑋) → ((𝑡𝑋 ∧ ∃𝑥𝐹 𝑥𝑡) → 𝑡𝐹))
103, 9sylbid 230 . . 3 (𝐹 ∈ (Fil‘𝑋) → (𝑡 ∈ (𝑋filGen𝐹) → 𝑡𝐹))
1110ssrdv 3594 . 2 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) ⊆ 𝐹)
12 ssfg 21616 . . 3 (𝐹 ∈ (fBas‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
131, 12syl 17 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ (𝑋filGen𝐹))
1411, 13eqssd 3605 1 (𝐹 ∈ (Fil‘𝑋) → (𝑋filGen𝐹) = 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wrex 2909  wss 3560  cfv 5857  (class class class)co 6615  fBascfbas 19674  filGencfg 19675  Filcfil 21589
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-pw 4138  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fv 5865  df-ov 6618  df-oprab 6619  df-mpt2 6620  df-fbas 19683  df-fg 19684  df-fil 21590
This theorem is referenced by:  elfilss  21620  fgtr  21634  fmid  21704  isfcf  21778  cnextcn  21811  filnetlem4  32071
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