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Theorem filss 21870
Description: A filter is closed under taking supersets. (Contributed by FL, 20-Jul-2007.) (Revised by Stefan O'Rear, 28-Jul-2015.)
Assertion
Ref Expression
filss ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → 𝐵𝐹)

Proof of Theorem filss
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 isfil 21864 . . . 4 (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
21simprbi 486 . . 3 (𝐹 ∈ (Fil‘𝑋) → ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹))
32adantr 468 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹))
4 elfvdm 6440 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil)
5 simp2 1160 . . 3 ((𝐴𝐹𝐵𝑋𝐴𝐵) → 𝐵𝑋)
6 elpw2g 5019 . . . 4 (𝑋 ∈ dom Fil → (𝐵 ∈ 𝒫 𝑋𝐵𝑋))
76biimpar 465 . . 3 ((𝑋 ∈ dom Fil ∧ 𝐵𝑋) → 𝐵 ∈ 𝒫 𝑋)
84, 5, 7syl2an 585 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → 𝐵 ∈ 𝒫 𝑋)
9 simpr1 1241 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → 𝐴𝐹)
10 simpr3 1245 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → 𝐴𝐵)
11 elpwg 4359 . . . . 5 (𝐴𝐹 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
129, 11syl 17 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
1310, 12mpbird 248 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → 𝐴 ∈ 𝒫 𝐵)
14 inelcm 4229 . . 3 ((𝐴𝐹𝐴 ∈ 𝒫 𝐵) → (𝐹 ∩ 𝒫 𝐵) ≠ ∅)
159, 13, 14syl2anc 575 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → (𝐹 ∩ 𝒫 𝐵) ≠ ∅)
16 pweq 4354 . . . . . 6 (𝑥 = 𝐵 → 𝒫 𝑥 = 𝒫 𝐵)
1716ineq2d 4013 . . . . 5 (𝑥 = 𝐵 → (𝐹 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝐵))
1817neeq1d 3037 . . . 4 (𝑥 = 𝐵 → ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝐵) ≠ ∅))
19 eleq1 2873 . . . 4 (𝑥 = 𝐵 → (𝑥𝐹𝐵𝐹))
2018, 19imbi12d 335 . . 3 (𝑥 = 𝐵 → (((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹) ↔ ((𝐹 ∩ 𝒫 𝐵) ≠ ∅ → 𝐵𝐹)))
2120rspccv 3499 . 2 (∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹) → (𝐵 ∈ 𝒫 𝑋 → ((𝐹 ∩ 𝒫 𝐵) ≠ ∅ → 𝐵𝐹)))
223, 8, 15, 21syl3c 66 1 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → 𝐵𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  w3a 1100   = wceq 1637  wcel 2156  wne 2978  wral 3096  cin 3768  wss 3769  c0 4116  𝒫 cpw 4351  dom cdm 5311  cfv 6101  fBascfbas 19942  Filcfil 21862
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-csb 3729  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-pw 4353  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fv 6109  df-fil 21863
This theorem is referenced by:  filin  21871  filtop  21872  isfil2  21873  infil  21880  fgfil  21892  fgabs  21896  filconn  21900  filuni  21902  trfil2  21904  trfg  21908  isufil2  21925  ufprim  21926  ufileu  21936  filufint  21937  elfm3  21967  rnelfm  21970  fmfnfmlem2  21972  fmfnfmlem4  21974  flimopn  21992  flimrest  22000  flimfnfcls  22045  fclscmpi  22046  alexsublem  22061  metust  22576  cfil3i  23279  cfilfcls  23284  iscmet3lem2  23302  equivcfil  23309  relcmpcmet  23327  minveclem4  23415  fgmin  32686
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