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Theorem ustfilxp 21227
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 5892 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
2 isust 21218 . . . . . . 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 245 . . . . 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 468 . . . 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 1020 . . 3  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  U  C_  ~P ( X  X.  X
) )
75simp2d 1021 . . . . 5  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  ( X  X.  X )  e.  U
)
8 ne0i 3737 . . . . 5  |-  ( ( X  X.  X )  e.  U  ->  U  =/=  (/) )
97, 8syl 17 . . . 4  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  U  =/=  (/) )
105simp3d 1022 . . . . . . . . . 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 2757 . . . . . . . . 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 1022 . . . . . . . 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 1020 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  (  _I  |`  X )  C_  v )
14 vex 3048 . . . . . . . . . . . . 13  |-  w  e. 
_V
15 opelresi 5116 . . . . . . . . . . . . 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 210 . . . . . . . . . . 11  |-  ( w  e.  X  ->  <. w ,  w >.  e.  (  _I  |`  X ) )
1817rgen 2747 . . . . . . . . . 10  |-  A. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X )
19 r19.2z 3858 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  A. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X )
)  ->  E. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X ) )
2018, 19mpan2 677 . . . . . . . . 9  |-  ( X  =/=  (/)  ->  E. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X ) )
2120ad2antrr 732 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  E. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X ) )
22 ne0i 3737 . . . . . . . . 9  |-  ( <.
w ,  w >.  e.  (  _I  |`  X )  ->  (  _I  |`  X )  =/=  (/) )
2322rexlimivw 2876 . . . . . . . 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 3767 . . . . . . 7  |-  ( ( (  _I  |`  X ) 
C_  v  /\  (  _I  |`  X )  =/=  (/) )  ->  v  =/=  (/) )
2613, 24, 25syl2anc 667 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  v  =/=  (/) )
2726nelrdva 3249 . . . . 5  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  -.  (/)  e.  U
)
28 df-nel 2625 . . . . 5  |-  ( (/)  e/  U  <->  -.  (/)  e.  U
)
2927, 28sylibr 216 . . . 4  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  (/)  e/  U
)
3011simp2d 1021 . . . . . . . . 9  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  A. w  e.  U  ( v  i^i  w )  e.  U
)
3130r19.21bi 2757 . . . . . . . 8  |-  ( ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  /\  v  e.  U )  /\  w  e.  U
)  ->  ( v  i^i  w )  e.  U
)
3214inex2 4545 . . . . . . . . . 10  |-  ( v  i^i  w )  e. 
_V
3332pwid 3965 . . . . . . . . 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 3618 . . . . . . 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 3737 . . . . . . 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 2802 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  A. w  e.  U  ( U  i^i  ~P ( v  i^i  w ) )  =/=  (/) )
3938ralrimiva 2802 . . . 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 1188 . . 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 6593 . . . . . 6  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
421, 1, 41syl2anc 667 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  e.  _V )
43 isfbas 20844 . . . . 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 468 . . 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 933 . 2  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  U  e.  ( fBas `  ( X  X.  X ) ) )
47 n0 3741 . . . . 5  |-  ( ( U  i^i  ~P w
)  =/=  (/)  <->  E. v 
v  e.  ( U  i^i  ~P w ) )
48 elin 3617 . . . . . . 7  |-  ( v  e.  ( U  i^i  ~P w )  <->  ( v  e.  U  /\  v  e.  ~P w ) )
49 selpw 3958 . . . . . . . 8  |-  ( v  e.  ~P w  <->  v  C_  w )
5049anbi2i 700 . . . . . . 7  |-  ( ( v  e.  U  /\  v  e.  ~P w
)  <->  ( v  e.  U  /\  v  C_  w ) )
5148, 50bitri 253 . . . . . 6  |-  ( v  e.  ( U  i^i  ~P w )  <->  ( v  e.  U  /\  v  C_  w ) )
5251exbii 1718 . . . . 5  |-  ( E. v  v  e.  ( U  i^i  ~P w
)  <->  E. v ( v  e.  U  /\  v  C_  w ) )
5347, 52bitri 253 . . . 4  |-  ( ( U  i^i  ~P w
)  =/=  (/)  <->  E. v
( v  e.  U  /\  v  C_  w ) )
5411simp1d 1020 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  A. w  e.  ~P  ( X  X.  X ) ( v 
C_  w  ->  w  e.  U ) )
5554r19.21bi 2757 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  /\  v  e.  U )  /\  w  e.  ~P ( X  X.  X
) )  ->  (
v  C_  w  ->  w  e.  U ) )
5655an32s 813 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  /\  w  e.  ~P ( X  X.  X ) )  /\  v  e.  U
)  ->  ( v  C_  w  ->  w  e.  U ) )
5756expimpd 608 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  w  e.  ~P ( X  X.  X
) )  ->  (
( v  e.  U  /\  v  C_  w )  ->  w  e.  U
) )
5857exlimdv 1779 . . . 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 221 . . 3  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  w  e.  ~P ( X  X.  X
) )  ->  (
( U  i^i  ~P w )  =/=  (/)  ->  w  e.  U ) )
6059ralrimiva 2802 . 2  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  A. w  e.  ~P  ( X  X.  X ) ( ( U  i^i  ~P w
)  =/=  (/)  ->  w  e.  U ) )
61 isfil 20862 . 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 670 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 188    /\ wa 371    /\ w3a 985   E.wex 1663    e. wcel 1887    =/= wne 2622    e/ wnel 2623   A.wral 2737   E.wrex 2738   _Vcvv 3045    i^i cin 3403    C_ wss 3404   (/)c0 3731   ~Pcpw 3951   <.cop 3974    _I cid 4744    X. cxp 4832   `'ccnv 4833    |` cres 4836    o. ccom 4838   ` cfv 5582   fBascfbas 18958   Filcfil 20860  UnifOncust 21214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-8 1889  ax-9 1896  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431  ax-sep 4525  ax-nul 4534  ax-pow 4581  ax-pr 4639  ax-un 6583
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 987  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2624  df-nel 2625  df-ral 2742  df-rex 2743  df-rab 2746  df-v 3047  df-sbc 3268  df-csb 3364  df-dif 3407  df-un 3409  df-in 3411  df-ss 3418  df-nul 3732  df-if 3882  df-pw 3953  df-sn 3969  df-pr 3971  df-op 3975  df-uni 4199  df-br 4403  df-opab 4462  df-mpt 4463  df-id 4749  df-xp 4840  df-rel 4841  df-cnv 4842  df-co 4843  df-dm 4844  df-rn 4845  df-res 4846  df-ima 4847  df-iota 5546  df-fun 5584  df-fv 5590  df-fbas 18967  df-fil 20861  df-ust 21215
This theorem is referenced by: (None)
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