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Theorem ustfilxp 20443
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 5884 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
2 isust 20434 . . . . . . 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 16 . . . . . 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 241 . . . . 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 466 . . . 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 1003 . . 3  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  U  C_  ~P ( X  X.  X
) )
75simp2d 1004 . . . . 5  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  ( X  X.  X )  e.  U
)
8 ne0i 3784 . . . . 5  |-  ( ( X  X.  X )  e.  U  ->  U  =/=  (/) )
97, 8syl 16 . . . 4  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  U  =/=  (/) )
105simp3d 1005 . . . . . . . . . 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 2826 . . . . . . . . 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 1005 . . . . . . . 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 1003 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  (  _I  |`  X )  C_  v )
14 vex 3109 . . . . . . . . . . . . 13  |-  w  e. 
_V
15 opelresi 5276 . . . . . . . . . . . . 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 206 . . . . . . . . . . 11  |-  ( w  e.  X  ->  <. w ,  w >.  e.  (  _I  |`  X ) )
1817rgen 2817 . . . . . . . . . 10  |-  A. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X )
19 r19.2z 3910 . . . . . . . . . 10  |-  ( ( X  =/=  (/)  /\  A. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X )
)  ->  E. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X ) )
2018, 19mpan2 671 . . . . . . . . 9  |-  ( X  =/=  (/)  ->  E. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X ) )
2120ad2antrr 725 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  E. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X ) )
22 ne0i 3784 . . . . . . . . 9  |-  ( <.
w ,  w >.  e.  (  _I  |`  X )  ->  (  _I  |`  X )  =/=  (/) )
2322rexlimivw 2945 . . . . . . . 8  |-  ( E. w  e.  X  <. w ,  w >.  e.  (  _I  |`  X )  ->  (  _I  |`  X )  =/=  (/) )
2421, 23syl 16 . . . . . . 7  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  (  _I  |`  X )  =/=  (/) )
25 ssn0 3811 . . . . . . 7  |-  ( ( (  _I  |`  X ) 
C_  v  /\  (  _I  |`  X )  =/=  (/) )  ->  v  =/=  (/) )
2613, 24, 25syl2anc 661 . . . . . 6  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  v  =/=  (/) )
2726nelrdva 3306 . . . . 5  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  -.  (/)  e.  U
)
28 df-nel 2658 . . . . 5  |-  ( (/)  e/  U  <->  -.  (/)  e.  U
)
2927, 28sylibr 212 . . . 4  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  (/)  e/  U
)
3011simp2d 1004 . . . . . . . . 9  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  A. w  e.  U  ( v  i^i  w )  e.  U
)
3130r19.21bi 2826 . . . . . . . 8  |-  ( ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  /\  v  e.  U )  /\  w  e.  U
)  ->  ( v  i^i  w )  e.  U
)
3214inex2 4582 . . . . . . . . . 10  |-  ( v  i^i  w )  e. 
_V
3332pwid 4017 . . . . . . . . 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 3681 . . . . . . 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 3784 . . . . . . 7  |-  ( ( v  i^i  w )  e.  ( U  i^i  ~P ( v  i^i  w
) )  ->  ( U  i^i  ~P ( v  i^i  w ) )  =/=  (/) )
3735, 36syl 16 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  /\  v  e.  U )  /\  w  e.  U
)  ->  ( U  i^i  ~P ( v  i^i  w ) )  =/=  (/) )
3837ralrimiva 2871 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  A. w  e.  U  ( U  i^i  ~P ( v  i^i  w ) )  =/=  (/) )
3938ralrimiva 2871 . . . 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 1171 . . 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 6702 . . . . . 6  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
421, 1, 41syl2anc 661 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  ( X  X.  X )  e.  _V )
43 isfbas 20058 . . . . 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 16 . . . 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 466 . . 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 915 . 2  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  U  e.  ( fBas `  ( X  X.  X ) ) )
47 n0 3787 . . . . 5  |-  ( ( U  i^i  ~P w
)  =/=  (/)  <->  E. v 
v  e.  ( U  i^i  ~P w ) )
48 elin 3680 . . . . . . 7  |-  ( v  e.  ( U  i^i  ~P w )  <->  ( v  e.  U  /\  v  e.  ~P w ) )
49 selpw 4010 . . . . . . . 8  |-  ( v  e.  ~P w  <->  v  C_  w )
5049anbi2i 694 . . . . . . 7  |-  ( ( v  e.  U  /\  v  e.  ~P w
)  <->  ( v  e.  U  /\  v  C_  w ) )
5148, 50bitri 249 . . . . . 6  |-  ( v  e.  ( U  i^i  ~P w )  <->  ( v  e.  U  /\  v  C_  w ) )
5251exbii 1639 . . . . 5  |-  ( E. v  v  e.  ( U  i^i  ~P w
)  <->  E. v ( v  e.  U  /\  v  C_  w ) )
5347, 52bitri 249 . . . 4  |-  ( ( U  i^i  ~P w
)  =/=  (/)  <->  E. v
( v  e.  U  /\  v  C_  w ) )
5411simp1d 1003 . . . . . . . 8  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  v  e.  U )  ->  A. w  e.  ~P  ( X  X.  X ) ( v 
C_  w  ->  w  e.  U ) )
5554r19.21bi 2826 . . . . . . 7  |-  ( ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  /\  v  e.  U )  /\  w  e.  ~P ( X  X.  X
) )  ->  (
v  C_  w  ->  w  e.  U ) )
5655an32s 802 . . . . . 6  |-  ( ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X ) )  /\  w  e.  ~P ( X  X.  X ) )  /\  v  e.  U
)  ->  ( v  C_  w  ->  w  e.  U ) )
5756expimpd 603 . . . . 5  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  w  e.  ~P ( X  X.  X
) )  ->  (
( v  e.  U  /\  v  C_  w )  ->  w  e.  U
) )
5857exlimdv 1695 . . . 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 217 . . 3  |-  ( ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  /\  w  e.  ~P ( X  X.  X
) )  ->  (
( U  i^i  ~P w )  =/=  (/)  ->  w  e.  U ) )
6059ralrimiva 2871 . 2  |-  ( ( X  =/=  (/)  /\  U  e.  (UnifOn `  X )
)  ->  A. w  e.  ~P  ( X  X.  X ) ( ( U  i^i  ~P w
)  =/=  (/)  ->  w  e.  U ) )
61 isfil 20076 . 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 664 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 184    /\ wa 369    /\ w3a 968   E.wex 1591    e. wcel 1762    =/= wne 2655    e/ wnel 2656   A.wral 2807   E.wrex 2808   _Vcvv 3106    i^i cin 3468    C_ wss 3469   (/)c0 3778   ~Pcpw 4003   <.cop 4026    _I cid 4783    X. cxp 4990   `'ccnv 4991    |` cres 4994    o. ccom 4996   ` cfv 5579   fBascfbas 18170   Filcfil 20074  UnifOncust 20430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-nel 2658  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fv 5587  df-fbas 18180  df-fil 20075  df-ust 20431
This theorem is referenced by: (None)
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