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Theorem ustssel 20535
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 996 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  U  e.  (UnifOn `  X )
)
21elfvexd 5894 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  X  e.  _V )
3 isust 20533 . . . . . 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 1010 . . 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 996 . . . 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 2857 . . 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 997 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  V  e.  U )
11 simp3 998 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  W  C_  ( X  X.  X
) )
12 xpexg 6587 . . . . . . 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 4594 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  W  C_  ( X  X.  X
) )  ->  W  e.  _V )
15 elpwg 4018 . . . . 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 3525 . . . . 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 3526 . . . . 5  |-  ( w  =  W  ->  ( V  C_  w  <->  V  C_  W
) )
21 eleq1 2539 . . . . 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 3223 . . 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 973    = wceq 1379    e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113    i^i cin 3475    C_ wss 3476   ~Pcpw 4010    _I cid 4790    X. cxp 4997   `'ccnv 4998    |` cres 5001    o. ccom 5003   ` cfv 5588  UnifOncust 20529
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-res 5011  df-iota 5551  df-fun 5590  df-fv 5596  df-ust 20530
This theorem is referenced by:  trust  20559  ustuqtop1  20571  ucnprima  20612
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