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Theorem ustinvel 21236
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 5897 . . . . . . 7  |-  ( U  e.  (UnifOn `  X
)  ->  X  e.  _V )
2 isust 21230 . . . . . . 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 17 . . . . . 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 245 . . . . 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 1023 . . . 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 3455 . . . . . . . 8  |-  ( v  =  V  ->  (
v  C_  w  <->  V  C_  w
) )
76imbi1d 319 . . . . . . 7  |-  ( v  =  V  ->  (
( v  C_  w  ->  w  e.  U )  <-> 
( V  C_  w  ->  w  e.  U ) ) )
87ralbidv 2829 . . . . . 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 3629 . . . . . . . 8  |-  ( v  =  V  ->  (
v  i^i  w )  =  ( V  i^i  w ) )
109eleq1d 2515 . . . . . . 7  |-  ( v  =  V  ->  (
( v  i^i  w
)  e.  U  <->  ( V  i^i  w )  e.  U
) )
1110ralbidv 2829 . . . . . 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 3456 . . . . . . 7  |-  ( v  =  V  ->  (
(  _I  |`  X ) 
C_  v  <->  (  _I  |`  X )  C_  V
) )
13 cnveq 5011 . . . . . . . 8  |-  ( v  =  V  ->  `' v  =  `' V
)
1413eleq1d 2515 . . . . . . 7  |-  ( v  =  V  ->  ( `' v  e.  U  <->  `' V  e.  U ) )
15 sseq2 3456 . . . . . . . 8  |-  ( v  =  V  ->  (
( w  o.  w
)  C_  v  <->  ( w  o.  w )  C_  V
) )
1615rexbidv 2903 . . . . . . 7  |-  ( v  =  V  ->  ( E. w  e.  U  ( w  o.  w
)  C_  v  <->  E. w  e.  U  ( w  o.  w )  C_  V
) )
1712, 14, 163anbi123d 1341 . . . . . 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 1341 . . . . 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 3148 . . . 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 472 . . 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 1023 . 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 1022 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  `' V  e.  U )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 986    = wceq 1446    e. wcel 1889   A.wral 2739   E.wrex 2740   _Vcvv 3047    i^i cin 3405    C_ wss 3406   ~Pcpw 3953    _I cid 4747    X. cxp 4835   `'ccnv 4836    |` cres 4839    o. ccom 4841   ` cfv 5585  UnifOncust 21226
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-ral 2744  df-rex 2745  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-res 4849  df-iota 5549  df-fun 5587  df-fv 5593  df-ust 21227
This theorem is referenced by:  ustexsym  21242  trust  21256
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