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Theorem ustdiag 19798
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 5732 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
2 isust 19793 . . . . . . 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 1002 . . . 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 3392 . . . . . . . 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 2750 . . . . . 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 3560 . . . . . . . 8  |-  ( v  =  V  ->  (
v  i^i  w )  =  ( V  i^i  w ) )
109eleq1d 2509 . . . . . . 7  |-  ( v  =  V  ->  (
( v  i^i  w
)  e.  U  <->  ( V  i^i  w )  e.  U
) )
1110ralbidv 2750 . . . . . 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 3393 . . . . . . 7  |-  ( v  =  V  ->  (
(  _I  |`  X ) 
C_  v  <->  (  _I  |`  X )  C_  V
) )
13 cnveq 5028 . . . . . . . 8  |-  ( v  =  V  ->  `' v  =  `' V
)
1413eleq1d 2509 . . . . . . 7  |-  ( v  =  V  ->  ( `' v  e.  U  <->  `' V  e.  U ) )
15 sseq2 3393 . . . . . . . 8  |-  ( v  =  V  ->  (
( w  o.  w
)  C_  v  <->  ( w  o.  w )  C_  V
) )
1615rexbidv 2751 . . . . . . 7  |-  ( v  =  V  ->  ( E. w  e.  U  ( w  o.  w
)  C_  v  <->  E. w  e.  U  ( w  o.  w )  C_  V
) )
1712, 14, 163anbi123d 1289 . . . . . 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 1289 . . . . 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 3084 . . . 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 1002 . 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 1000 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 965    = wceq 1369    e. wcel 1756   A.wral 2730   E.wrex 2731   _Vcvv 2987    i^i cin 3342    C_ wss 3343   ~Pcpw 3875    _I cid 4646    X. cxp 4853   `'ccnv 4854    |` cres 4857    o. ccom 4859   ` cfv 5433  UnifOncust 19789
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 2423  ax-sep 4428  ax-nul 4436  ax-pow 4485  ax-pr 4546  ax-un 6387
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 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2577  df-ne 2622  df-ral 2735  df-rex 2736  df-rab 2739  df-v 2989  df-sbc 3202  df-csb 3304  df-dif 3346  df-un 3348  df-in 3350  df-ss 3357  df-nul 3653  df-if 3807  df-pw 3877  df-sn 3893  df-pr 3895  df-op 3899  df-uni 4107  df-br 4308  df-opab 4366  df-mpt 4367  df-id 4651  df-xp 4861  df-rel 4862  df-cnv 4863  df-co 4864  df-dm 4865  df-res 4867  df-iota 5396  df-fun 5435  df-fv 5441  df-ust 19790
This theorem is referenced by:  ustssco  19804  ustref  19808  ustelimasn  19812  trust  19819  ustuqtop3  19833
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