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Theorem ustneism 19798
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 3992 . . . 4  |-  ( A  e.  X  ->  { A }  =/=  (/) )
21adantl 466 . . 3  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  { A }  =/=  (/) )
3 xpco 5377 . . 3  |-  ( { A }  =/=  (/)  ->  (
( { A }  X.  ( V " { A } ) )  o.  ( ( V " { A } )  X. 
{ A } ) )  =  ( ( V " { A } )  X.  ( V " { A }
) ) )
42, 3syl 16 . 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 5255 . . . . 5  |-  `' ( { A }  X.  ( V " { A } ) )  =  ( ( V " { A } )  X. 
{ A } )
6 ressn 5373 . . . . . . 7  |-  ( V  |`  { A } )  =  ( { A }  X.  ( V " { A } ) )
76cnveqi 5014 . . . . . 6  |-  `' ( V  |`  { A } )  =  `' ( { A }  X.  ( V " { A } ) )
8 resss 5134 . . . . . . 7  |-  ( V  |`  { A } ) 
C_  V
9 cnvss 5012 . . . . . . 7  |-  ( ( V  |`  { A } )  C_  V  ->  `' ( V  |`  { A } )  C_  `' V )
108, 9ax-mp 5 . . . . . 6  |-  `' ( V  |`  { A } )  C_  `' V
117, 10eqsstr3i 3387 . . . . 5  |-  `' ( { A }  X.  ( V " { A } ) )  C_  `' V
125, 11eqsstr3i 3387 . . . 4  |-  ( ( V " { A } )  X.  { A } )  C_  `' V
13 coss2 4996 . . . 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 12 . . 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 3387 . . . 4  |-  ( { A }  X.  ( V " { A }
) )  C_  V
16 coss1 4995 . . . 4  |-  ( ( { A }  X.  ( V " { A } ) )  C_  V  ->  ( ( { A }  X.  ( V " { A }
) )  o.  `' V )  C_  ( V  o.  `' V
) )
1715, 16mp1i 12 . . 3  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  (
( { A }  X.  ( V " { A } ) )  o.  `' V )  C_  ( V  o.  `' V
) )
1814, 17sstrd 3366 . 2  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  (
( { A }  X.  ( V " { A } ) )  o.  ( ( V " { A } )  X. 
{ A } ) )  C_  ( V  o.  `' V ) )
194, 18eqsstr3d 3391 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 369    = wceq 1369    e. wcel 1756    =/= wne 2606    C_ wss 3328   (/)c0 3637   {csn 3877    X. cxp 4838   `'ccnv 4839    |` cres 4842   "cima 4843    o. ccom 4844
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-br 4293  df-opab 4351  df-xp 4846  df-rel 4847  df-cnv 4848  df-co 4849  df-dm 4850  df-rn 4851  df-res 4852  df-ima 4853
This theorem is referenced by:  neipcfilu  19871
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