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Theorem filunibas 21732
Description: Recover the base set from a filter. (Contributed by Stefan O'Rear, 2-Aug-2015.)
Assertion
Ref Expression
filunibas (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)

Proof of Theorem filunibas
StepHypRef Expression
1 filsspw 21702 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
2 sspwuni 4643 . . 3 (𝐹 ⊆ 𝒫 𝑋 𝐹𝑋)
31, 2sylib 208 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐹𝑋)
4 filtop 21706 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
5 unissel 4500 . 2 (( 𝐹𝑋𝑋𝐹) → 𝐹 = 𝑋)
63, 4, 5syl2anc 694 1 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wss 3607  𝒫 cpw 4191   cuni 4468  cfv 5926  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-nel 2927  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-fbas 19791  df-fil 21697
This theorem is referenced by:  filunirn  21733  filconn  21734  uffixfr  21774  uffix2  21775  uffixsn  21776  ufildr  21782  flimtopon  21821  flimss1  21824  flffval  21840  fclsval  21859  isfcls  21860  fclstopon  21863  fclsfnflim  21878  fcfval  21884
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