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Theorem utopsnnei 21312
Description: Images of singletons by entourages  V are neighborhoods of those singletons. (Contributed by Thierry Arnoux, 13-Jan-2018.)
Hypothesis
Ref Expression
utoptop.1  |-  J  =  (unifTop `  U )
Assertion
Ref Expression
utopsnnei  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  ( V " { P }
)  e.  ( ( nei `  J ) `
 { P }
) )

Proof of Theorem utopsnnei
Dummy variable  v is distinct from all other variables.
StepHypRef Expression
1 eqid 2461 . . . 4  |-  ( V
" { P }
)  =  ( V
" { P }
)
2 imaeq1 5181 . . . . . 6  |-  ( v  =  V  ->  (
v " { P } )  =  ( V " { P } ) )
32eqeq2d 2471 . . . . 5  |-  ( v  =  V  ->  (
( V " { P } )  =  ( v " { P } )  <->  ( V " { P } )  =  ( V " { P } ) ) )
43rspcev 3161 . . . 4  |-  ( ( V  e.  U  /\  ( V " { P } )  =  ( V " { P } ) )  ->  E. v  e.  U  ( V " { P } )  =  ( v " { P } ) )
51, 4mpan2 682 . . 3  |-  ( V  e.  U  ->  E. v  e.  U  ( V " { P } )  =  ( v " { P } ) )
653ad2ant2 1036 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  E. v  e.  U  ( V " { P } )  =  ( v " { P } ) )
7 utoptop.1 . . . . . 6  |-  J  =  (unifTop `  U )
87utopsnneip 21311 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  J
) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
983adant2 1033 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  (
( nei `  J
) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
109eleq2d 2524 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  (
( V " { P } )  e.  ( ( nei `  J
) `  { P } )  <->  ( V " { P } )  e.  ran  ( v  e.  U  |->  ( v
" { P }
) ) ) )
11 imaexg 6756 . . . . 5  |-  ( V  e.  U  ->  ( V " { P }
)  e.  _V )
12 eqid 2461 . . . . . 6  |-  ( v  e.  U  |->  ( v
" { P }
) )  =  ( v  e.  U  |->  ( v " { P } ) )
1312elrnmpt 5099 . . . . 5  |-  ( ( V " { P } )  e.  _V  ->  ( ( V " { P } )  e. 
ran  ( v  e.  U  |->  ( v " { P } ) )  <->  E. v  e.  U  ( V " { P } )  =  ( v " { P } ) ) )
1411, 13syl 17 . . . 4  |-  ( V  e.  U  ->  (
( V " { P } )  e.  ran  ( v  e.  U  |->  ( v " { P } ) )  <->  E. v  e.  U  ( V " { P } )  =  ( v " { P } ) ) )
15143ad2ant2 1036 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  (
( V " { P } )  e.  ran  ( v  e.  U  |->  ( v " { P } ) )  <->  E. v  e.  U  ( V " { P } )  =  ( v " { P } ) ) )
1610, 15bitrd 261 . 2  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  (
( V " { P } )  e.  ( ( nei `  J
) `  { P } )  <->  E. v  e.  U  ( V " { P } )  =  ( v " { P } ) ) )
176, 16mpbird 240 1  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  ( V " { P }
)  e.  ( ( nei `  J ) `
 { P }
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    /\ w3a 991    = wceq 1454    e. wcel 1897   E.wrex 2749   _Vcvv 3056   {csn 3979    |-> cmpt 4474   ran crn 4853   "cima 4855   ` cfv 5600   neicnei 20161  UnifOncust 21262  unifTopcutop 21293
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-8 1899  ax-9 1906  ax-10 1925  ax-11 1930  ax-12 1943  ax-13 2101  ax-ext 2441  ax-rep 4528  ax-sep 4538  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6609
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 992  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-reu 2755  df-rab 2757  df-v 3058  df-sbc 3279  df-csb 3375  df-dif 3418  df-un 3420  df-in 3422  df-ss 3429  df-pss 3431  df-nul 3743  df-if 3893  df-pw 3964  df-sn 3980  df-pr 3982  df-tp 3984  df-op 3986  df-uni 4212  df-int 4248  df-iun 4293  df-br 4416  df-opab 4475  df-mpt 4476  df-tr 4511  df-eprel 4763  df-id 4767  df-po 4773  df-so 4774  df-fr 4811  df-we 4813  df-xp 4858  df-rel 4859  df-cnv 4860  df-co 4861  df-dm 4862  df-rn 4863  df-res 4864  df-ima 4865  df-pred 5398  df-ord 5444  df-on 5445  df-lim 5446  df-suc 5447  df-iota 5564  df-fun 5602  df-fn 5603  df-f 5604  df-f1 5605  df-fo 5606  df-f1o 5607  df-fv 5608  df-ov 6317  df-oprab 6318  df-mpt2 6319  df-om 6719  df-wrecs 7053  df-recs 7115  df-rdg 7153  df-1o 7207  df-oadd 7211  df-er 7388  df-en 7595  df-fin 7598  df-fi 7950  df-top 19969  df-nei 20162  df-ust 21263  df-utop 21294
This theorem is referenced by:  utop2nei  21313  utop3cls  21314  utopreg  21315
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