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Theorem filunibas 21894
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 21864 . . 3 (𝐹 ∈ (Fil‘𝑋) → 𝐹 ⊆ 𝒫 𝑋)
2 sspwuni 4799 . . 3 (𝐹 ⊆ 𝒫 𝑋 𝐹𝑋)
31, 2sylib 209 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝐹𝑋)
4 filtop 21868 . 2 (𝐹 ∈ (Fil‘𝑋) → 𝑋𝐹)
5 unissel 4658 . 2 (( 𝐹𝑋𝑋𝐹) → 𝐹 = 𝑋)
63, 4, 5syl2anc 575 1 (𝐹 ∈ (Fil‘𝑋) → 𝐹 = 𝑋)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1637  wcel 2158  wss 3766  𝒫 cpw 4348   cuni 4626  cfv 6098  Filcfil 21858
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1880  ax-4 1897  ax-5 2004  ax-6 2070  ax-7 2106  ax-8 2160  ax-9 2167  ax-10 2187  ax-11 2203  ax-12 2216  ax-13 2422  ax-ext 2784  ax-sep 4971  ax-nul 4980  ax-pow 5032  ax-pr 5093
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 1865  df-sb 2063  df-eu 2636  df-mo 2637  df-clab 2792  df-cleq 2798  df-clel 2801  df-nfc 2936  df-ne 2978  df-nel 3081  df-ral 3100  df-rex 3101  df-rab 3104  df-v 3392  df-sbc 3631  df-csb 3726  df-dif 3769  df-un 3771  df-in 3773  df-ss 3780  df-nul 4114  df-if 4277  df-pw 4350  df-sn 4368  df-pr 4370  df-op 4374  df-uni 4627  df-br 4841  df-opab 4903  df-mpt 4920  df-id 5216  df-xp 5314  df-rel 5315  df-cnv 5316  df-co 5317  df-dm 5318  df-rn 5319  df-res 5320  df-ima 5321  df-iota 6061  df-fun 6100  df-fv 6106  df-fbas 19947  df-fil 21859
This theorem is referenced by:  filunirn  21895  filconn  21896  uffixfr  21936  uffix2  21937  uffixsn  21938  ufildr  21944  flimtopon  21983  flimss1  21986  flffval  22002  fclsval  22021  isfcls  22022  fclstopon  22025  fclsfnflim  22040  fcfval  22046
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