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Theorem utopsnnei 19836
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 2443 . . . 4  |-  ( V
" { P }
)  =  ( V
" { P }
)
2 imaeq1 5176 . . . . . 6  |-  ( v  =  V  ->  (
v " { P } )  =  ( V " { P } ) )
32eqeq2d 2454 . . . . 5  |-  ( v  =  V  ->  (
( V " { P } )  =  ( v " { P } )  <->  ( V " { P } )  =  ( V " { P } ) ) )
43rspcev 3085 . . . 4  |-  ( ( V  e.  U  /\  ( V " { P } )  =  ( V " { P } ) )  ->  E. v  e.  U  ( V " { P } )  =  ( v " { P } ) )
51, 4mpan2 671 . . 3  |-  ( V  e.  U  ->  E. v  e.  U  ( V " { P } )  =  ( v " { P } ) )
653ad2ant2 1010 . 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 19835 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  J
) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
983adant2 1007 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  (
( nei `  J
) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
109eleq2d 2510 . . 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 6527 . . . . 5  |-  ( V  e.  U  ->  ( V " { P }
)  e.  _V )
12 eqid 2443 . . . . . 6  |-  ( v  e.  U  |->  ( v
" { P }
) )  =  ( v  e.  U  |->  ( v " { P } ) )
1312elrnmpt 5098 . . . . 5  |-  ( ( V " { P } )  e.  _V  ->  ( ( V " { P } )  e. 
ran  ( v  e.  U  |->  ( v " { P } ) )  <->  E. v  e.  U  ( V " { P } )  =  ( v " { P } ) ) )
1411, 13syl 16 . . . 4  |-  ( V  e.  U  ->  (
( V " { P } )  e.  ran  ( v  e.  U  |->  ( v " { P } ) )  <->  E. v  e.  U  ( V " { P } )  =  ( v " { P } ) ) )
15143ad2ant2 1010 . . 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 253 . 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 232 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 184    /\ w3a 965    = wceq 1369    e. wcel 1756   E.wrex 2728   _Vcvv 2984   {csn 3889    e. cmpt 4362   ran crn 4853   "cima 4855   ` cfv 5430   neicnei 18713  UnifOncust 19786  unifTopcutop 19817
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-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-rep 4415  ax-sep 4425  ax-nul 4433  ax-pow 4482  ax-pr 4543  ax-un 6384
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  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 2577  df-ne 2620  df-ral 2732  df-rex 2733  df-reu 2734  df-rab 2736  df-v 2986  df-sbc 3199  df-csb 3301  df-dif 3343  df-un 3345  df-in 3347  df-ss 3354  df-pss 3356  df-nul 3650  df-if 3804  df-pw 3874  df-sn 3890  df-pr 3892  df-tp 3894  df-op 3896  df-uni 4104  df-int 4141  df-iun 4185  df-br 4305  df-opab 4363  df-mpt 4364  df-tr 4398  df-eprel 4644  df-id 4648  df-po 4653  df-so 4654  df-fr 4691  df-we 4693  df-ord 4734  df-on 4735  df-lim 4736  df-suc 4737  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-iota 5393  df-fun 5432  df-fn 5433  df-f 5434  df-f1 5435  df-fo 5436  df-f1o 5437  df-fv 5438  df-ov 6106  df-oprab 6107  df-mpt2 6108  df-om 6489  df-recs 6844  df-rdg 6878  df-1o 6932  df-oadd 6936  df-er 7113  df-en 7323  df-fin 7326  df-fi 7673  df-top 18515  df-nei 18714  df-ust 19787  df-utop 19818
This theorem is referenced by:  utop2nei  19837  utop3cls  19838  utopreg  19839
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