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Theorem ustneism 21230
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 4115 . . . 4  |-  ( A  e.  X  ->  { A }  =/=  (/) )
21adantl 468 . . 3  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  { A }  =/=  (/) )
3 xpco 5393 . . 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 5271 . . . . 5  |-  `' ( { A }  X.  ( V " { A } ) )  =  ( ( V " { A } )  X. 
{ A } )
6 ressn 5389 . . . . . . 7  |-  ( V  |`  { A } )  =  ( { A }  X.  ( V " { A } ) )
76cnveqi 5026 . . . . . 6  |-  `' ( V  |`  { A } )  =  `' ( { A }  X.  ( V " { A } ) )
8 resss 5145 . . . . . . 7  |-  ( V  |`  { A } ) 
C_  V
9 cnvss 5024 . . . . . . 7  |-  ( ( V  |`  { A } )  C_  V  ->  `' ( V  |`  { A } )  C_  `' V )
108, 9ax-mp 5 . . . . . 6  |-  `' ( V  |`  { A } )  C_  `' V
117, 10eqsstr3i 3496 . . . . 5  |-  `' ( { A }  X.  ( V " { A } ) )  C_  `' V
125, 11eqsstr3i 3496 . . . 4  |-  ( ( V " { A } )  X.  { A } )  C_  `' V
13 coss2 5008 . . . 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 3496 . . . 4  |-  ( { A }  X.  ( V " { A }
) )  C_  V
16 coss1 5007 . . . 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 3475 . 2  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  (
( { A }  X.  ( V " { A } ) )  o.  ( ( V " { A } )  X. 
{ A } ) )  C_  ( V  o.  `' V ) )
194, 18eqsstr3d 3500 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 371    = wceq 1438    e. wcel 1869    =/= wne 2619    C_ wss 3437   (/)c0 3762   {csn 3997    X. cxp 4849   `'ccnv 4850    |` cres 4853   "cima 4854    o. ccom 4855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-opab 4481  df-xp 4857  df-rel 4858  df-cnv 4859  df-co 4860  df-dm 4861  df-rn 4862  df-res 4863  df-ima 4864
This theorem is referenced by:  neipcfilu  21303
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