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Theorem ustneism 20477
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 4144 . . . 4  |-  ( A  e.  X  ->  { A }  =/=  (/) )
21adantl 466 . . 3  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  { A }  =/=  (/) )
3 xpco 5546 . . 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 5423 . . . . 5  |-  `' ( { A }  X.  ( V " { A } ) )  =  ( ( V " { A } )  X. 
{ A } )
6 ressn 5542 . . . . . . 7  |-  ( V  |`  { A } )  =  ( { A }  X.  ( V " { A } ) )
76cnveqi 5176 . . . . . 6  |-  `' ( V  |`  { A } )  =  `' ( { A }  X.  ( V " { A } ) )
8 resss 5296 . . . . . . 7  |-  ( V  |`  { A } ) 
C_  V
9 cnvss 5174 . . . . . . 7  |-  ( ( V  |`  { A } )  C_  V  ->  `' ( V  |`  { A } )  C_  `' V )
108, 9ax-mp 5 . . . . . 6  |-  `' ( V  |`  { A } )  C_  `' V
117, 10eqsstr3i 3535 . . . . 5  |-  `' ( { A }  X.  ( V " { A } ) )  C_  `' V
125, 11eqsstr3i 3535 . . . 4  |-  ( ( V " { A } )  X.  { A } )  C_  `' V
13 coss2 5158 . . . 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 3535 . . . 4  |-  ( { A }  X.  ( V " { A }
) )  C_  V
16 coss1 5157 . . . 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 3514 . 2  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  (
( { A }  X.  ( V " { A } ) )  o.  ( ( V " { A } )  X. 
{ A } ) )  C_  ( V  o.  `' V ) )
194, 18eqsstr3d 3539 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 1379    e. wcel 1767    =/= wne 2662    C_ wss 3476   (/)c0 3785   {csn 4027    X. cxp 4997   `'ccnv 4998    |` cres 5001   "cima 5002    o. ccom 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012
This theorem is referenced by:  neipcfilu  20550
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