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Theorem utopsnnei 20618
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 2467 . . . 4  |-  ( V
" { P }
)  =  ( V
" { P }
)
2 imaeq1 5338 . . . . . 6  |-  ( v  =  V  ->  (
v " { P } )  =  ( V " { P } ) )
32eqeq2d 2481 . . . . 5  |-  ( v  =  V  ->  (
( V " { P } )  =  ( v " { P } )  <->  ( V " { P } )  =  ( V " { P } ) ) )
43rspcev 3219 . . . 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 1018 . 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 20617 . . . . 5  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  J
) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
983adant2 1015 . . . 4  |-  ( ( U  e.  (UnifOn `  X )  /\  V  e.  U  /\  P  e.  X )  ->  (
( nei `  J
) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
109eleq2d 2537 . . 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 6732 . . . . 5  |-  ( V  e.  U  ->  ( V " { P }
)  e.  _V )
12 eqid 2467 . . . . . 6  |-  ( v  e.  U  |->  ( v
" { P }
) )  =  ( v  e.  U  |->  ( v " { P } ) )
1312elrnmpt 5255 . . . . 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 1018 . . 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 973    = wceq 1379    e. wcel 1767   E.wrex 2818   _Vcvv 3118   {csn 4033    |-> cmpt 4511   ran crn 5006   "cima 5008   ` cfv 5594   neicnei 19464  UnifOncust 20568  unifTopcutop 20599
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-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-rep 4564  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692  ax-un 6587
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  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 2822  df-rex 2823  df-reu 2824  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-tp 4038  df-op 4040  df-uni 4252  df-int 4289  df-iun 4333  df-br 4454  df-opab 4512  df-mpt 4513  df-tr 4547  df-eprel 4797  df-id 4801  df-po 4806  df-so 4807  df-fr 4844  df-we 4846  df-ord 4887  df-on 4888  df-lim 4889  df-suc 4890  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fn 5597  df-f 5598  df-f1 5599  df-fo 5600  df-f1o 5601  df-fv 5602  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-om 6696  df-recs 7054  df-rdg 7088  df-1o 7142  df-oadd 7146  df-er 7323  df-en 7529  df-fin 7532  df-fi 7883  df-top 19266  df-nei 19465  df-ust 20569  df-utop 20600
This theorem is referenced by:  utop2nei  20619  utop3cls  20620  utopreg  20621
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