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Theorem ustneism 21286
Description: For a point  A in  X,  ( V " { A } ) is small enough in  ( V  o.  `' V ). This proposition actually does not require any axiom of the definition of uniform structures. (Contributed by Thierry Arnoux, 18-Nov-2017.)
Assertion
Ref Expression
ustneism  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  (
( V " { A } )  X.  ( V " { A }
) )  C_  ( V  o.  `' V
) )

Proof of Theorem ustneism
StepHypRef Expression
1 snnzg 4101 . . . 4  |-  ( A  e.  X  ->  { A }  =/=  (/) )
21adantl 472 . . 3  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  { A }  =/=  (/) )
3 xpco 5394 . . 3  |-  ( { A }  =/=  (/)  ->  (
( { A }  X.  ( V " { A } ) )  o.  ( ( V " { A } )  X. 
{ A } ) )  =  ( ( V " { A } )  X.  ( V " { A }
) ) )
42, 3syl 17 . 2  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  (
( { A }  X.  ( V " { A } ) )  o.  ( ( V " { A } )  X. 
{ A } ) )  =  ( ( V " { A } )  X.  ( V " { A }
) ) )
5 cnvxp 5272 . . . . 5  |-  `' ( { A }  X.  ( V " { A } ) )  =  ( ( V " { A } )  X. 
{ A } )
6 ressn 5390 . . . . . . 7  |-  ( V  |`  { A } )  =  ( { A }  X.  ( V " { A } ) )
76cnveqi 5027 . . . . . 6  |-  `' ( V  |`  { A } )  =  `' ( { A }  X.  ( V " { A } ) )
8 resss 5146 . . . . . . 7  |-  ( V  |`  { A } ) 
C_  V
9 cnvss 5025 . . . . . . 7  |-  ( ( V  |`  { A } )  C_  V  ->  `' ( V  |`  { A } )  C_  `' V )
108, 9ax-mp 5 . . . . . 6  |-  `' ( V  |`  { A } )  C_  `' V
117, 10eqsstr3i 3474 . . . . 5  |-  `' ( { A }  X.  ( V " { A } ) )  C_  `' V
125, 11eqsstr3i 3474 . . . 4  |-  ( ( V " { A } )  X.  { A } )  C_  `' V
13 coss2 5009 . . . 4  |-  ( ( ( V " { A } )  X.  { A } )  C_  `' V  ->  ( ( { A }  X.  ( V " { A }
) )  o.  (
( V " { A } )  X.  { A } ) )  C_  ( ( { A }  X.  ( V " { A } ) )  o.  `' V ) )
1412, 13mp1i 13 . . 3  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  (
( { A }  X.  ( V " { A } ) )  o.  ( ( V " { A } )  X. 
{ A } ) )  C_  ( ( { A }  X.  ( V " { A }
) )  o.  `' V ) )
156, 8eqsstr3i 3474 . . . 4  |-  ( { A }  X.  ( V " { A }
) )  C_  V
16 coss1 5008 . . . 4  |-  ( ( { A }  X.  ( V " { A } ) )  C_  V  ->  ( ( { A }  X.  ( V " { A }
) )  o.  `' V )  C_  ( V  o.  `' V
) )
1715, 16mp1i 13 . . 3  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  (
( { A }  X.  ( V " { A } ) )  o.  `' V )  C_  ( V  o.  `' V
) )
1814, 17sstrd 3453 . 2  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  (
( { A }  X.  ( V " { A } ) )  o.  ( ( V " { A } )  X. 
{ A } ) )  C_  ( V  o.  `' V ) )
194, 18eqsstr3d 3478 1  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  (
( V " { A } )  X.  ( V " { A }
) )  C_  ( V  o.  `' V
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    = wceq 1454    e. wcel 1897    =/= wne 2632    C_ wss 3415   (/)c0 3742   {csn 3979    X. cxp 4850   `'ccnv 4851    |` cres 4854   "cima 4855    o. ccom 4856
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1679  ax-4 1692  ax-5 1768  ax-6 1815  ax-7 1861  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-sep 4538  ax-nul 4547  ax-pr 4652
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 993  df-tru 1457  df-ex 1674  df-nf 1678  df-sb 1808  df-eu 2313  df-mo 2314  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2591  df-ne 2634  df-ral 2753  df-rex 2754  df-rab 2757  df-v 3058  df-sbc 3279  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-nul 3743  df-if 3893  df-sn 3980  df-pr 3982  df-op 3986  df-br 4416  df-opab 4475  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865
This theorem is referenced by:  neipcfilu  21359
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