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Theorem ustinvel 20838
Description: If  V is an entourage, so is its inverse. Condition UII of [BourbakiTop1] p. II.1. (Contributed by Thierry Arnoux, 2-Dec-2017.)
Assertion
Ref Expression
ustinvel  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  `' V  e.  U )

Proof of Theorem ustinvel
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 20832 . . . . . . 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 3520 . . . . . . . 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 2896 . . . . . 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 3689 . . . . . . . 8  |-  ( v  =  V  ->  (
v  i^i  w )  =  ( V  i^i  w ) )
109eleq1d 2526 . . . . . . 7  |-  ( v  =  V  ->  (
( v  i^i  w
)  e.  U  <->  ( V  i^i  w )  e.  U
) )
1110ralbidv 2896 . . . . . 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 3521 . . . . . . 7  |-  ( v  =  V  ->  (
(  _I  |`  X ) 
C_  v  <->  (  _I  |`  X )  C_  V
) )
13 cnveq 5186 . . . . . . . 8  |-  ( v  =  V  ->  `' v  =  `' V
)
1413eleq1d 2526 . . . . . . 7  |-  ( v  =  V  ->  ( `' v  e.  U  <->  `' V  e.  U ) )
15 sseq2 3521 . . . . . . . 8  |-  ( v  =  V  ->  (
( w  o.  w
)  C_  v  <->  ( w  o.  w )  C_  V
) )
1615rexbidv 2968 . . . . . . 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 3206 . . . 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 ) )
2221simp2d 1009 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  `' V  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    /\ w3a 973    = wceq 1395    e. wcel 1819   A.wral 2807   E.wrex 2808   _Vcvv 3109    i^i cin 3470    C_ wss 3471   ~Pcpw 4015    _I cid 4799    X. cxp 5006   `'ccnv 5007    |` cres 5010    o. ccom 5012   ` cfv 5594  UnifOncust 20828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-mpt 4517  df-id 4804  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-res 5020  df-iota 5557  df-fun 5596  df-fv 5602  df-ust 20829
This theorem is referenced by:  ustexsym  20844  trust  20858
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