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Theorem filss 21704
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 21698 . . . 4 (𝐹 ∈ (Fil‘𝑋) ↔ (𝐹 ∈ (fBas‘𝑋) ∧ ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹)))
21simprbi 479 . . 3 (𝐹 ∈ (Fil‘𝑋) → ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹))
32adantr 480 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → ∀𝑥 ∈ 𝒫 𝑋((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹))
4 elfvdm 6258 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝑋 ∈ dom Fil)
5 simp2 1082 . . 3 ((𝐴𝐹𝐵𝑋𝐴𝐵) → 𝐵𝑋)
6 elpw2g 4857 . . . 4 (𝑋 ∈ dom Fil → (𝐵 ∈ 𝒫 𝑋𝐵𝑋))
76biimpar 501 . . 3 ((𝑋 ∈ dom Fil ∧ 𝐵𝑋) → 𝐵 ∈ 𝒫 𝑋)
84, 5, 7syl2an 493 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → 𝐵 ∈ 𝒫 𝑋)
9 simpr1 1087 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → 𝐴𝐹)
10 simpr3 1089 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → 𝐴𝐵)
11 elpwg 4199 . . . . 5 (𝐴𝐹 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
129, 11syl 17 . . . 4 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
1310, 12mpbird 247 . . 3 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → 𝐴 ∈ 𝒫 𝐵)
14 inelcm 4065 . . 3 ((𝐴𝐹𝐴 ∈ 𝒫 𝐵) → (𝐹 ∩ 𝒫 𝐵) ≠ ∅)
159, 13, 14syl2anc 694 . 2 ((𝐹 ∈ (Fil‘𝑋) ∧ (𝐴𝐹𝐵𝑋𝐴𝐵)) → (𝐹 ∩ 𝒫 𝐵) ≠ ∅)
16 pweq 4194 . . . . . 6 (𝑥 = 𝐵 → 𝒫 𝑥 = 𝒫 𝐵)
1716ineq2d 3847 . . . . 5 (𝑥 = 𝐵 → (𝐹 ∩ 𝒫 𝑥) = (𝐹 ∩ 𝒫 𝐵))
1817neeq1d 2882 . . . 4 (𝑥 = 𝐵 → ((𝐹 ∩ 𝒫 𝑥) ≠ ∅ ↔ (𝐹 ∩ 𝒫 𝐵) ≠ ∅))
19 eleq1 2718 . . . 4 (𝑥 = 𝐵 → (𝑥𝐹𝐵𝐹))
2018, 19imbi12d 333 . . 3 (𝑥 = 𝐵 → (((𝐹 ∩ 𝒫 𝑥) ≠ ∅ → 𝑥𝐹) ↔ ((𝐹 ∩ 𝒫 𝐵) ≠ ∅ → 𝐵𝐹)))
2120rspccv 3337 . 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 196  wa 383  w3a 1054   = wceq 1523  wcel 2030  wne 2823  wral 2941  cin 3606  wss 3607  c0 3948  𝒫 cpw 4191  dom cdm 5143  cfv 5926  fBascfbas 19782  Filcfil 21696
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fv 5934  df-fil 21697
This theorem is referenced by:  filin  21705  filtop  21706  isfil2  21707  infil  21714  fgfil  21726  fgabs  21730  filconn  21734  filuni  21736  trfil2  21738  trfg  21742  isufil2  21759  ufprim  21760  ufileu  21770  filufint  21771  elfm3  21801  rnelfm  21804  fmfnfmlem2  21806  fmfnfmlem4  21808  flimopn  21826  flimrest  21834  flimfnfcls  21879  fclscmpi  21880  alexsublem  21895  metust  22410  cfil3i  23113  cfilfcls  23118  iscmet3lem2  23136  equivcfil  23143  relcmpcmet  23161  minveclem4  23249  fgmin  32490
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