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Theorem ustfilxp 21305
Description: A uniform structure on a nonempty base is a filter. Remark 3 of [BourbakiTop1] p. II.2. (Contributed by Thierry Arnoux, 15-Nov-2017.)
Assertion
Ref Expression
ustfilxp  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  U  e.  ( Fil `  ( X  X.  X ) ) )

Proof of Theorem ustfilxp
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5906 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
2 isust 21296 . . . . . . 7  |-  ( X  e.  _V  ->  ( U  e.  (UnifOn `  X
)  <->  ( U  C_  ~P ( X  X.  X
)  /\  ( X  X.  X )  e.  U  /\  A. v  e.  U  ( A. w  e.  ~P  ( X  X.  X
) ( v  C_  w  ->  w  e.  U
)  /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  ( (  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v ) ) ) ) )
31, 2syl 17 . . . . . 6  |-  ( U  e.  (UnifOn `  X
)  ->  ( U  e.  (UnifOn `  X )  <->  ( U  C_  ~P ( X  X.  X )  /\  ( X  X.  X
)  e.  U  /\  A. v  e.  U  ( A. w  e.  ~P  ( X  X.  X
) ( v  C_  w  ->  w  e.  U
)  /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  ( (  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v ) ) ) ) )
43ibi 249 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( U  C_ 
~P ( X  X.  X )  /\  ( X  X.  X )  e.  U  /\  A. v  e.  U  ( A. w  e.  ~P  ( X  X.  X ) ( v  C_  w  ->  w  e.  U )  /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  (
(  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v ) ) ) )
54adantl 473 . . . 4  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  ( U  C_ 
~P ( X  X.  X )  /\  ( X  X.  X )  e.  U  /\  A. v  e.  U  ( A. w  e.  ~P  ( X  X.  X ) ( v  C_  w  ->  w  e.  U )  /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  (
(  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v ) ) ) )
65simp1d 1042 . . 3  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  U  C_  ~P ( X  X.  X
) )
75simp2d 1043 . . . . 5  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  ( X  X.  X )  e.  U
)
8 ne0i 3728 . . . . 5  |-  ( ( X  X.  X )  e.  U  ->  U  =/=  (/) )
97, 8syl 17 . . . 4  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  U  =/=  (/) )
105simp3d 1044 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  A. v  e.  U  ( A. w  e.  ~P  ( X  X.  X ) ( v  C_  w  ->  w  e.  U )  /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  (
(  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v ) ) )
1110r19.21bi 2776 . . . . . . . . 9  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  ( A. w  e.  ~P  ( X  X.  X
) ( v  C_  w  ->  w  e.  U
)  /\  A. w  e.  U  ( v  i^i  w )  e.  U  /\  ( (  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v ) ) )
1211simp3d 1044 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  (
(  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v ) )
1312simp1d 1042 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  (  _I  |`  X )  C_  v )
14 vex 3034 . . . . . . . . . . . . 13  |-  w  e. 
_V
15 opelresi 5122 . . . . . . . . . . . . 13  |-  ( w  e.  _V  ->  ( <. w ,  w >.  e.  (  _I  |`  X )  <-> 
w  e.  X ) )
1614, 15ax-mp 5 . . . . . . . . . . . 12  |-  ( <.
w ,  w >.  e.  (  _I  |`  X )  <-> 
w  e.  X )
1716biimpri 211 . . . . . . . . . . 11  |-  ( w  e.  X  ->  <. w ,  w >.  e.  (  _I  |`  X ) )
1817rgen 2766 . . . . . . . . . 10  |-  A. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X )
19 r19.2z 3849 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  A. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X )
)  ->  E. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X ) )
2018, 19mpan2 685 . . . . . . . . 9  |-  ( X  =/=  (/)  ->  E. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X ) )
2120ad2antrr 740 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  E. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X ) )
22 ne0i 3728 . . . . . . . . 9  |-  ( <.
w ,  w >.  e.  (  _I  |`  X )  ->  (  _I  |`  X )  =/=  (/) )
2322rexlimivw 2869 . . . . . . . 8  |-  ( E. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X )  ->  (  _I  |`  X )  =/=  (/) )
2421, 23syl 17 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  (  _I  |`  X )  =/=  (/) )
25 ssn0 3770 . . . . . . 7  |-  ( ( (  _I  |`  X ) 
C_  v  /\  (  _I  |`  X )  =/=  (/) )  ->  v  =/=  (/) )
2613, 24, 25syl2anc 673 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  v  =/=  (/) )
2726nelrdva 3237 . . . . 5  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  -.  (/)  e.  U
)
28 df-nel 2644 . . . . 5  |-  ( (/)  e/  U  <->  -.  (/)  e.  U
)
2927, 28sylibr 217 . . . 4  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  (/)  e/  U
)
3011simp2d 1043 . . . . . . . . 9  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  A. w  e.  U  ( v  i^i  w )  e.  U
)
3130r19.21bi 2776 . . . . . . . 8  |-  ( ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  /\  v  e.  U )  /\  w  e.  U
)  ->  ( v  i^i  w )  e.  U
)
3214inex2 4538 . . . . . . . . . 10  |-  ( v  i^i  w )  e. 
_V
3332pwid 3956 . . . . . . . . 9  |-  ( v  i^i  w )  e. 
~P ( v  i^i  w )
3433a1i 11 . . . . . . . 8  |-  ( ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  /\  v  e.  U )  /\  w  e.  U
)  ->  ( v  i^i  w )  e.  ~P ( v  i^i  w
) )
3531, 34elind 3609 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  /\  v  e.  U )  /\  w  e.  U
)  ->  ( v  i^i  w )  e.  ( U  i^i  ~P (
v  i^i  w )
) )
36 ne0i 3728 . . . . . . 7  |-  ( ( v  i^i  w )  e.  ( U  i^i  ~P ( v  i^i  w
) )  ->  ( U  i^i  ~P ( v  i^i  w ) )  =/=  (/) )
3735, 36syl 17 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  /\  v  e.  U )  /\  w  e.  U
)  ->  ( U  i^i  ~P ( v  i^i  w ) )  =/=  (/) )
3837ralrimiva 2809 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  A. w  e.  U  ( U  i^i  ~P ( v  i^i  w ) )  =/=  (/) )
3938ralrimiva 2809 . . . 4  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  A. v  e.  U  A. w  e.  U  ( U  i^i  ~P ( v  i^i  w ) )  =/=  (/) )
409, 29, 393jca 1210 . . 3  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  ( U  =/=  (/)  /\  (/)  e/  U  /\  A. v  e.  U  A. w  e.  U  ( U  i^i  ~P (
v  i^i  w )
)  =/=  (/) ) )
41 xpexg 6612 . . . . . 6  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
421, 1, 41syl2anc 673 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  e.  _V )
43 isfbas 20922 . . . . 5  |-  ( ( X  X.  X )  e.  _V  ->  ( U  e.  ( fBas `  ( X  X.  X
) )  <->  ( U  C_ 
~P ( X  X.  X )  /\  ( U  =/=  (/)  /\  (/)  e/  U  /\  A. v  e.  U  A. w  e.  U  ( U  i^i  ~P (
v  i^i  w )
)  =/=  (/) ) ) ) )
4442, 43syl 17 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( U  e.  ( fBas `  ( X  X.  X ) )  <-> 
( U  C_  ~P ( X  X.  X
)  /\  ( U  =/=  (/)  /\  (/)  e/  U  /\  A. v  e.  U  A. w  e.  U  ( U  i^i  ~P (
v  i^i  w )
)  =/=  (/) ) ) ) )
4544adantl 473 . . 3  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  ( U  e.  ( fBas `  ( X  X.  X ) )  <-> 
( U  C_  ~P ( X  X.  X
)  /\  ( U  =/=  (/)  /\  (/)  e/  U  /\  A. v  e.  U  A. w  e.  U  ( U  i^i  ~P (
v  i^i  w )
)  =/=  (/) ) ) ) )
466, 40, 45mpbir2and 936 . 2  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  U  e.  ( fBas `  ( X  X.  X ) ) )
47 n0 3732 . . . . 5  |-  ( ( U  i^i  ~P w
)  =/=  (/)  <->  E. v 
v  e.  ( U  i^i  ~P w ) )
48 elin 3608 . . . . . . 7  |-  ( v  e.  ( U  i^i  ~P w )  <->  ( v  e.  U  /\  v  e.  ~P w ) )
49 selpw 3949 . . . . . . . 8  |-  ( v  e.  ~P w  <->  v  C_  w )
5049anbi2i 708 . . . . . . 7  |-  ( ( v  e.  U  /\  v  e.  ~P w
)  <->  ( v  e.  U  /\  v  C_  w ) )
5148, 50bitri 257 . . . . . 6  |-  ( v  e.  ( U  i^i  ~P w )  <->  ( v  e.  U  /\  v  C_  w ) )
5251exbii 1726 . . . . 5  |-  ( E. v  v  e.  ( U  i^i  ~P w
)  <->  E. v ( v  e.  U  /\  v  C_  w ) )
5347, 52bitri 257 . . . 4  |-  ( ( U  i^i  ~P w
)  =/=  (/)  <->  E. v
( v  e.  U  /\  v  C_  w ) )
5411simp1d 1042 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  A. w  e.  ~P  ( X  X.  X ) ( v 
C_  w  ->  w  e.  U ) )
5554r19.21bi 2776 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  /\  v  e.  U )  /\  w  e.  ~P ( X  X.  X
) )  ->  (
v  C_  w  ->  w  e.  U ) )
5655an32s 821 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  /\  w  e.  ~P ( X  X.  X ) )  /\  v  e.  U
)  ->  ( v  C_  w  ->  w  e.  U ) )
5756expimpd 614 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  w  e.  ~P ( X  X.  X
) )  ->  (
( v  e.  U  /\  v  C_  w )  ->  w  e.  U
) )
5857exlimdv 1787 . . . 4  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  w  e.  ~P ( X  X.  X
) )  ->  ( E. v ( v  e.  U  /\  v  C_  w )  ->  w  e.  U ) )
5953, 58syl5bi 225 . . 3  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  w  e.  ~P ( X  X.  X
) )  ->  (
( U  i^i  ~P w )  =/=  (/)  ->  w  e.  U ) )
6059ralrimiva 2809 . 2  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  A. w  e.  ~P  ( X  X.  X ) ( ( U  i^i  ~P w
)  =/=  (/)  ->  w  e.  U ) )
61 isfil 20940 . 2  |-  ( U  e.  ( Fil `  ( X  X.  X ) )  <-> 
( U  e.  (
fBas `  ( X  X.  X ) )  /\  A. w  e.  ~P  ( X  X.  X ) ( ( U  i^i  ~P w )  =/=  (/)  ->  w  e.  U ) ) )
6246, 60, 61sylanbrc 677 1  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  U  e.  ( Fil `  ( X  X.  X ) ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 189    /\ wa 376    /\ w3a 1007   E.wex 1671    e. wcel 1904    =/= wne 2641    e/ wnel 2642   A.wral 2756   E.wrex 2757   _Vcvv 3031    i^i cin 3389    C_ wss 3390   (/)c0 3722   ~Pcpw 3942   <.cop 3965    _I cid 4749    X. cxp 4837   `'ccnv 4838    |` cres 4841    o. ccom 4843   ` cfv 5589   fBascfbas 19035   Filcfil 20938  UnifOncust 21292
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fv 5597  df-fbas 19044  df-fil 20939  df-ust 21293
This theorem is referenced by: (None)
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