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Theorem ustssel 21151
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 1005 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  U  e.  (UnifOn `  X )
)
21elfvexd 5909 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  X  e.  _V )
3 isust 21149 . . . . . 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 17 . . . . 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 213 . . . 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 1019 . . 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 1005 . . . 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 2825 . . 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 17 . 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 1006 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  V  e.  U )
11 simp3 1007 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  W  C_  ( X  X.  X
) )
12 xpexg 6607 . . . . . . 7  |-  ( ( X  e.  _V  /\  X  e.  _V )  ->  ( X  X.  X
)  e.  _V )
132, 2, 12syl2anc 665 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  ( X  X.  X )  e. 
_V )
1413, 11ssexd 4572 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  W  e.  _V )
15 elpwg 3993 . . . . 5  |-  ( W  e.  _V  ->  ( W  e.  ~P ( X  X.  X )  <->  W  C_  ( X  X.  X ) ) )
1614, 15syl 17 . . . 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 235 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  W  e.  ~P ( X  X.  X ) )
18 sseq1 3491 . . . . 5  |-  ( v  =  V  ->  (
v  C_  w  <->  V  C_  w
) )
1918imbi1d 318 . . . 4  |-  ( v  =  V  ->  (
( v  C_  w  ->  w  e.  U )  <-> 
( V  C_  w  ->  w  e.  U ) ) )
20 sseq2 3492 . . . . 5  |-  ( w  =  W  ->  ( V  C_  w  <->  V  C_  W
) )
21 eleq1 2501 . . . . 5  |-  ( w  =  W  ->  (
w  e.  U  <->  W  e.  U ) )
2220, 21imbi12d 321 . . . 4  |-  ( w  =  W  ->  (
( V  C_  w  ->  w  e.  U )  <-> 
( V  C_  W  ->  W  e.  U ) ) )
2319, 22rspc2v 3197 . . 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 665 . 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 187    /\ w3a 982    = wceq 1437    e. wcel 1870   A.wral 2782   E.wrex 2783   _Vcvv 3087    i^i cin 3441    C_ wss 3442   ~Pcpw 3985    _I cid 4764    X. cxp 4852   `'ccnv 4853    |` cres 4856    o. ccom 4858   ` cfv 5601  UnifOncust 21145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-res 4866  df-iota 5565  df-fun 5603  df-fv 5609  df-ust 21146
This theorem is referenced by:  trust  21175  ustuqtop1  21187  ucnprima  21228
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