MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ustneism Structured version   Unicode version

Theorem ustneism 20892
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 4133 . . . 4  |-  ( A  e.  X  ->  { A }  =/=  (/) )
21adantl 464 . . 3  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  { A }  =/=  (/) )
3 xpco 5530 . . 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 5409 . . . . 5  |-  `' ( { A }  X.  ( V " { A } ) )  =  ( ( V " { A } )  X. 
{ A } )
6 ressn 5526 . . . . . . 7  |-  ( V  |`  { A } )  =  ( { A }  X.  ( V " { A } ) )
76cnveqi 5166 . . . . . 6  |-  `' ( V  |`  { A } )  =  `' ( { A }  X.  ( V " { A } ) )
8 resss 5285 . . . . . . 7  |-  ( V  |`  { A } ) 
C_  V
9 cnvss 5164 . . . . . . 7  |-  ( ( V  |`  { A } )  C_  V  ->  `' ( V  |`  { A } )  C_  `' V )
108, 9ax-mp 5 . . . . . 6  |-  `' ( V  |`  { A } )  C_  `' V
117, 10eqsstr3i 3520 . . . . 5  |-  `' ( { A }  X.  ( V " { A } ) )  C_  `' V
125, 11eqsstr3i 3520 . . . 4  |-  ( ( V " { A } )  X.  { A } )  C_  `' V
13 coss2 5148 . . . 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 3520 . . . 4  |-  ( { A }  X.  ( V " { A }
) )  C_  V
16 coss1 5147 . . . 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 3499 . 2  |-  ( ( V  C_  ( X  X.  X )  /\  A  e.  X )  ->  (
( { A }  X.  ( V " { A } ) )  o.  ( ( V " { A } )  X. 
{ A } ) )  C_  ( V  o.  `' V ) )
194, 18eqsstr3d 3524 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 367    = wceq 1398    e. wcel 1823    =/= wne 2649    C_ wss 3461   (/)c0 3783   {csn 4016    X. cxp 4986   `'ccnv 4987    |` cres 4990   "cima 4991    o. ccom 4992
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001
This theorem is referenced by:  neipcfilu  20965
  Copyright terms: Public domain W3C validator