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Theorem ustssel 19749
Description: A uniform structure is upward closed. Condition FI of [BourbakiTop1] p. I.36. (Contributed by Thierry Arnoux, 19-Nov-2017.)
Assertion
Ref Expression
ustssel  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  ( V  C_  W  ->  W  e.  U ) )

Proof of Theorem ustssel
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simp1 988 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  U  e.  (UnifOn `  X )
)
21elfvexd 5711 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  X  e.  _V )
3 isust 19747 . . . . . 6  |-  ( 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 ) ) ) ) )
42, 3syl 16 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  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 ) ) ) ) )
51, 4mpbid 210 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  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 ) ) ) )
65simp3d 1002 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  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 ) ) )
7 simp1 988 . . . 4  |-  ( ( 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 ) )  ->  A. w  e.  ~P  ( X  X.  X
) ( v  C_  w  ->  w  e.  U
) )
87ralimi 2785 . . 3  |-  ( 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 ) )  ->  A. v  e.  U  A. w  e.  ~P  ( X  X.  X
) ( v  C_  w  ->  w  e.  U
) )
96, 8syl 16 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  A. v  e.  U  A. w  e.  ~P  ( X  X.  X ) ( v 
C_  w  ->  w  e.  U ) )
10 simp2 989 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  V  e.  U )
11 simp3 990 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  W  C_  ( X  X.  X
) )
12 xpexg 6502 . . . . . . 7  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
132, 2, 12syl2anc 661 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  ( X  X.  X )  e. 
_V )
1413, 11ssexd 4432 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  W  e.  _V )
15 elpwg 3861 . . . . 5  |-  ( W  e.  _V  ->  ( W  e.  ~P ( X  X.  X )  <->  W  C_  ( X  X.  X ) ) )
1614, 15syl 16 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  ( W  e.  ~P ( X  X.  X )  <->  W  C_  ( X  X.  X ) ) )
1711, 16mpbird 232 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  W  e.  ~P ( X  X.  X ) )
18 sseq1 3370 . . . . 5  |-  ( v  =  V  ->  (
v  C_  w  <->  V  C_  w
) )
1918imbi1d 317 . . . 4  |-  ( v  =  V  ->  (
( v  C_  w  ->  w  e.  U )  <-> 
( V  C_  w  ->  w  e.  U ) ) )
20 sseq2 3371 . . . . 5  |-  ( w  =  W  ->  ( V  C_  w  <->  V  C_  W
) )
21 eleq1 2497 . . . . 5  |-  ( w  =  W  ->  (
w  e.  U  <->  W  e.  U ) )
2220, 21imbi12d 320 . . . 4  |-  ( w  =  W  ->  (
( V  C_  w  ->  w  e.  U )  <-> 
( V  C_  W  ->  W  e.  U ) ) )
2319, 22rspc2v 3072 . . 3  |-  ( ( V  e.  U  /\  W  e.  ~P ( X  X.  X ) )  ->  ( A. v  e.  U  A. w  e.  ~P  ( X  X.  X ) ( v 
C_  w  ->  w  e.  U )  ->  ( V  C_  W  ->  W  e.  U ) ) )
2410, 17, 23syl2anc 661 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  ( A. v  e.  U  A. w  e.  ~P  ( X  X.  X
) ( v  C_  w  ->  w  e.  U
)  ->  ( V  C_  W  ->  W  e.  U ) ) )
259, 24mpd 15 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  ( V  C_  W  ->  W  e.  U ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ w3a 965    = wceq 1369    e. wcel 1756   A.wral 2709   E.wrex 2710   _Vcvv 2966    i^i cin 3320    C_ wss 3321   ~Pcpw 3853    _I cid 4623    X. cxp 4830   `'ccnv 4831    |` cres 4834    o. ccom 4836   ` cfv 5411  UnifOncust 19743
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2418  ax-sep 4406  ax-nul 4414  ax-pow 4463  ax-pr 4524  ax-un 6367
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2714  df-rex 2715  df-rab 2718  df-v 2968  df-sbc 3180  df-csb 3282  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3631  df-if 3785  df-pw 3855  df-sn 3871  df-pr 3873  df-op 3877  df-uni 4085  df-br 4286  df-opab 4344  df-mpt 4345  df-id 4628  df-xp 4838  df-rel 4839  df-cnv 4840  df-co 4841  df-dm 4842  df-res 4844  df-iota 5374  df-fun 5413  df-fv 5419  df-ust 19744
This theorem is referenced by:  trust  19773  ustuqtop1  19785  ucnprima  19826
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