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Theorem ustex2sym 21162
Description: In an uniform structure, for any entourage  V, there exists a symmetrical entourage smaller than half  V. (Contributed by Thierry Arnoux, 16-Jan-2018.)
Assertion
Ref Expression
ustex2sym  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  w
)  C_  V )
)
Distinct variable groups:    w, U    w, V    w, X

Proof of Theorem ustex2sym
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 simplll 766 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  ->  U  e.  (UnifOn `  X
) )
2 simplr 760 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  -> 
v  e.  U )
3 ustexsym 21161 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  v  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  v ) )
41, 2, 3syl2anc 665 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  v ) )
5 simprl 762 . . . . . 6  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  v ) )  ->  `' w  =  w
)
6 coss1 5010 . . . . . . . . 9  |-  ( w 
C_  v  ->  (
w  o.  w ) 
C_  ( v  o.  w ) )
7 coss2 5011 . . . . . . . . 9  |-  ( w 
C_  v  ->  (
v  o.  w ) 
C_  ( v  o.  v ) )
86, 7sstrd 3480 . . . . . . . 8  |-  ( w 
C_  v  ->  (
w  o.  w ) 
C_  ( v  o.  v ) )
98ad2antll 733 . . . . . . 7  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  v ) )  -> 
( w  o.  w
)  C_  ( v  o.  v ) )
10 simpllr 767 . . . . . . 7  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  v ) )  -> 
( v  o.  v
)  C_  V )
119, 10sstrd 3480 . . . . . 6  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  v ) )  -> 
( w  o.  w
)  C_  V )
125, 11jca 534 . . . . 5  |-  ( ( ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  /\  ( `' w  =  w  /\  w  C_  v ) )  -> 
( `' w  =  w  /\  ( w  o.  w )  C_  V ) )
1312ex 435 . . . 4  |-  ( ( ( ( ( U  e.  (UnifOn `  X
)  /\  V  e.  U )  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  /\  w  e.  U )  ->  ( ( `' w  =  w  /\  w  C_  v )  ->  ( `' w  =  w  /\  ( w  o.  w
)  C_  V )
) )
1413reximdva 2907 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  -> 
( E. w  e.  U  ( `' w  =  w  /\  w  C_  v )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  w
)  C_  V )
) )
154, 14mpd 15 . 2  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  v  e.  U )  /\  (
v  o.  v ) 
C_  V )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  w
)  C_  V )
)
16 ustexhalf 21156 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. v  e.  U  ( v  o.  v )  C_  V
)
1715, 16r19.29a 2977 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  ( w  o.  w
)  C_  V )
)
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    = wceq 1437    e. wcel 1870   E.wrex 2783    C_ wss 3442   `'ccnv 4853    o. ccom 4858   ` cfv 5601  UnifOncust 21145
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-8 1872  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pow 4603  ax-pr 4661  ax-un 6597
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-csb 3402  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-pw 3987  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-mpt 4486  df-id 4769  df-xp 4860  df-rel 4861  df-cnv 4862  df-co 4863  df-dm 4864  df-rn 4865  df-res 4866  df-iota 5565  df-fun 5603  df-fv 5609  df-ust 21146
This theorem is referenced by:  ustex3sym  21163
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