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Theorem ustexsym 19903
Description: In an uniform structure, for any entourage  V, there exists a smaller symmetrical entourage. (Contributed by Thierry Arnoux, 4-Jan-2018.)
Assertion
Ref Expression
ustexsym  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  V ) )
Distinct variable groups:    w, U    w, V
Allowed substitution hint:    X( w)

Proof of Theorem ustexsym
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 simplll 757 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  U  e.  (UnifOn `  X
) )
2 simplr 754 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  x  e.  U )
3 ustinvel 19897 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  `' x  e.  U )
41, 2, 3syl2anc 661 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  `' x  e.  U
)
5 ustincl 19895 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  `' x  e.  U  /\  x  e.  U )  ->  ( `' x  i^i  x )  e.  U
)
61, 4, 2, 5syl3anc 1219 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  -> 
( `' x  i^i  x )  e.  U
)
7 ustrel 19899 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  Rel  x )
8 dfrel2 5383 . . . . . . 7  |-  ( Rel  x  <->  `' `' x  =  x
)
97, 8sylib 196 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  `' `' x  =  x
)
109ineq1d 3646 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  ( `' `' x  i^i  `' x
)  =  ( x  i^i  `' x ) )
11 cnvin 5339 . . . . 5  |-  `' ( `' x  i^i  x
)  =  ( `' `' x  i^i  `' x
)
12 incom 3638 . . . . 5  |-  ( `' x  i^i  x )  =  ( x  i^i  `' x )
1310, 11, 123eqtr4g 2516 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  `' ( `' x  i^i  x
)  =  ( `' x  i^i  x ) )
141, 2, 13syl2anc 661 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  `' ( `' x  i^i  x )  =  ( `' x  i^i  x
) )
15 inss2 3666 . . . 4  |-  ( `' x  i^i  x ) 
C_  x
16 ustssco 19902 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  x  e.  U )  ->  x  C_  ( x  o.  x
) )
171, 2, 16syl2anc 661 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  x  C_  ( x  o.  x ) )
18 simpr 461 . . . . 5  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  -> 
( x  o.  x
)  C_  V )
1917, 18sstrd 3461 . . . 4  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  x  C_  V )
2015, 19syl5ss 3462 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  -> 
( `' x  i^i  x )  C_  V
)
21 cnveq 5108 . . . . . 6  |-  ( w  =  ( `' x  i^i  x )  ->  `' w  =  `' ( `' x  i^i  x
) )
22 id 22 . . . . . 6  |-  ( w  =  ( `' x  i^i  x )  ->  w  =  ( `' x  i^i  x ) )
2321, 22eqeq12d 2472 . . . . 5  |-  ( w  =  ( `' x  i^i  x )  ->  ( `' w  =  w  <->  `' ( `' x  i^i  x )  =  ( `' x  i^i  x
) ) )
24 sseq1 3472 . . . . 5  |-  ( w  =  ( `' x  i^i  x )  ->  (
w  C_  V  <->  ( `' x  i^i  x )  C_  V ) )
2523, 24anbi12d 710 . . . 4  |-  ( w  =  ( `' x  i^i  x )  ->  (
( `' w  =  w  /\  w  C_  V )  <->  ( `' ( `' x  i^i  x
)  =  ( `' x  i^i  x )  /\  ( `' x  i^i  x )  C_  V
) ) )
2625rspcev 3166 . . 3  |-  ( ( ( `' x  i^i  x )  e.  U  /\  ( `' ( `' x  i^i  x )  =  ( `' x  i^i  x )  /\  ( `' x  i^i  x
)  C_  V )
)  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  V ) )
276, 14, 20, 26syl12anc 1217 . 2  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  V  e.  U
)  /\  x  e.  U )  /\  (
x  o.  x ) 
C_  V )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  V ) )
28 ustexhalf 19898 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. x  e.  U  ( x  o.  x )  C_  V
)
2927, 28r19.29a 2955 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  E. w  e.  U  ( `' w  =  w  /\  w  C_  V ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   E.wrex 2794    i^i cin 3422    C_ wss 3423   `'ccnv 4934    o. ccom 4939   Rel wrel 4940   ` cfv 5513  UnifOncust 19887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4508  ax-nul 4516  ax-pow 4565  ax-pr 4626  ax-un 6469
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2599  df-ne 2644  df-ral 2798  df-rex 2799  df-rab 2802  df-v 3067  df-sbc 3282  df-csb 3384  df-dif 3426  df-un 3428  df-in 3430  df-ss 3437  df-nul 3733  df-if 3887  df-pw 3957  df-sn 3973  df-pr 3975  df-op 3979  df-uni 4187  df-br 4388  df-opab 4446  df-mpt 4447  df-id 4731  df-xp 4941  df-rel 4942  df-cnv 4943  df-co 4944  df-dm 4945  df-rn 4946  df-res 4947  df-iota 5476  df-fun 5515  df-fv 5521  df-ust 19888
This theorem is referenced by:  ustex2sym  19904  neipcfilu  19984
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