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Theorem ustdiag 20579
Description: The diagonal set is included in any entourage, i.e. any point is  V -close to itself. Condition UI of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
Assertion
Ref Expression
ustdiag  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (  _I  |`  X )  C_  V )

Proof of Theorem ustdiag
Dummy variables  v  w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elfvex 5899 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
2 isust 20574 . . . . . . 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 ) ) ) )
54simp3d 1010 . . . 4  |-  ( 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 ) ) )
6 sseq1 3530 . . . . . . . 8  |-  ( v  =  V  ->  (
v  C_  w  <->  V  C_  w
) )
76imbi1d 317 . . . . . . 7  |-  ( v  =  V  ->  (
( v  C_  w  ->  w  e.  U )  <-> 
( V  C_  w  ->  w  e.  U ) ) )
87ralbidv 2906 . . . . . 6  |-  ( v  =  V  ->  ( A. w  e.  ~P  ( X  X.  X
) ( v  C_  w  ->  w  e.  U
)  <->  A. w  e.  ~P  ( X  X.  X
) ( V  C_  w  ->  w  e.  U
) ) )
9 ineq1 3698 . . . . . . . 8  |-  ( v  =  V  ->  (
v  i^i  w )  =  ( V  i^i  w ) )
109eleq1d 2536 . . . . . . 7  |-  ( v  =  V  ->  (
( v  i^i  w
)  e.  U  <->  ( V  i^i  w )  e.  U
) )
1110ralbidv 2906 . . . . . 6  |-  ( v  =  V  ->  ( A. w  e.  U  ( v  i^i  w
)  e.  U  <->  A. w  e.  U  ( V  i^i  w )  e.  U
) )
12 sseq2 3531 . . . . . . 7  |-  ( v  =  V  ->  (
(  _I  |`  X ) 
C_  v  <->  (  _I  |`  X )  C_  V
) )
13 cnveq 5182 . . . . . . . 8  |-  ( v  =  V  ->  `' v  =  `' V
)
1413eleq1d 2536 . . . . . . 7  |-  ( v  =  V  ->  ( `' v  e.  U  <->  `' V  e.  U ) )
15 sseq2 3531 . . . . . . . 8  |-  ( v  =  V  ->  (
( w  o.  w
)  C_  v  <->  ( w  o.  w )  C_  V
) )
1615rexbidv 2978 . . . . . . 7  |-  ( v  =  V  ->  ( E. w  e.  U  ( w  o.  w
)  C_  v  <->  E. w  e.  U  ( w  o.  w )  C_  V
) )
1712, 14, 163anbi123d 1299 . . . . . 6  |-  ( v  =  V  ->  (
( (  _I  |`  X ) 
C_  v  /\  `' v  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  v )  <->  ( (  _I  |`  X )  C_  V  /\  `' V  e.  U  /\  E. w  e.  U  ( w  o.  w )  C_  V
) ) )
188, 11, 173anbi123d 1299 . . . . 5  |-  ( v  =  V  ->  (
( 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 )  /\  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
) ) ) )
1918rspcv 3215 . . . 4  |-  ( V  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 ) )  ->  ( 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
) ) ) )
205, 19mpan9 469 . . 3  |-  ( ( 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 ) ) )
2120simp3d 1010 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (
(  _I  |`  X ) 
C_  V  /\  `' V  e.  U  /\  E. w  e.  U  ( w  o.  w ) 
C_  V ) )
2221simp1d 1008 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (  _I  |`  X )  C_  V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767   A.wral 2817   E.wrex 2818   _Vcvv 3118    i^i cin 3480    C_ wss 3481   ~Pcpw 4016    _I cid 4796    X. cxp 5003   `'ccnv 5004    |` cres 5007    o. ccom 5009   ` cfv 5594  UnifOncust 20570
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 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
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 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-res 5017  df-iota 5557  df-fun 5596  df-fv 5602  df-ust 20571
This theorem is referenced by:  ustssco  20585  ustref  20589  ustelimasn  20593  trust  20600  ustuqtop3  20614
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