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Theorem ustssco 19924
Description: In an uniform structure, any entourage  V is a subset of its composition with itself. (Contributed by Thierry Arnoux, 5-Jan-2018.)
Assertion
Ref Expression
ustssco  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( V  o.  V
) )

Proof of Theorem ustssco
StepHypRef Expression
1 ssun1 3630 . . . 4  |-  V  C_  ( V  u.  ( V  o.  V )
)
2 coires1 5466 . . . . . 6  |-  ( V  o.  (  _I  |`  X ) )  =  ( V  |`  X )
3 ustrel 19921 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  Rel  V )
4 ustssxp 19914 . . . . . . . . 9  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( X  X.  X
) )
5 dmss 5150 . . . . . . . . 9  |-  ( V 
C_  ( X  X.  X )  ->  dom  V 
C_  dom  ( X  X.  X ) )
64, 5syl 16 . . . . . . . 8  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  dom  V 
C_  dom  ( X  X.  X ) )
7 dmxpid 5170 . . . . . . . 8  |-  dom  ( X  X.  X )  =  X
86, 7syl6sseq 3513 . . . . . . 7  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  dom  V 
C_  X )
9 relssres 5258 . . . . . . 7  |-  ( ( Rel  V  /\  dom  V 
C_  X )  -> 
( V  |`  X )  =  V )
103, 8, 9syl2anc 661 . . . . . 6  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  ( V  |`  X )  =  V )
112, 10syl5eq 2507 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  ( V  o.  (  _I  |`  X ) )  =  V )
1211uneq1d 3620 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (
( V  o.  (  _I  |`  X ) )  u.  ( V  o.  V ) )  =  ( V  u.  ( V  o.  V )
) )
131, 12syl5sseqr 3516 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( ( V  o.  (  _I  |`  X ) )  u.  ( V  o.  V ) ) )
14 coundi 5450 . . 3  |-  ( V  o.  ( (  _I  |`  X )  u.  V
) )  =  ( ( V  o.  (  _I  |`  X ) )  u.  ( V  o.  V ) )
1513, 14syl6sseqr 3514 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( V  o.  (
(  _I  |`  X )  u.  V ) ) )
16 ustdiag 19918 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (  _I  |`  X )  C_  V )
17 ssequn1 3637 . . . 4  |-  ( (  _I  |`  X )  C_  V  <->  ( (  _I  |`  X )  u.  V
)  =  V )
1816, 17sylib 196 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  (
(  _I  |`  X )  u.  V )  =  V )
1918coeq2d 5113 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  ( V  o.  ( (  _I  |`  X )  u.  V ) )  =  ( V  o.  V
) )
2015, 19sseqtrd 3503 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U )  ->  V  C_  ( V  o.  V
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758    u. cun 3437    C_ wss 3439    _I cid 4742    X. cxp 4949   dom cdm 4951    |` cres 4953    o. ccom 4955   Rel wrel 4956   ` cfv 5529  UnifOncust 19909
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 1955  ax-ext 2432  ax-sep 4524  ax-nul 4532  ax-pow 4581  ax-pr 4642  ax-un 6485
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 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-csb 3399  df-dif 3442  df-un 3444  df-in 3446  df-ss 3453  df-nul 3749  df-if 3903  df-pw 3973  df-sn 3989  df-pr 3991  df-op 3995  df-uni 4203  df-br 4404  df-opab 4462  df-mpt 4463  df-id 4747  df-xp 4957  df-rel 4958  df-cnv 4959  df-co 4960  df-dm 4961  df-rn 4962  df-res 4963  df-iota 5492  df-fun 5531  df-fv 5537  df-ust 19910
This theorem is referenced by:  ustexsym  19925  ustex3sym  19927
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