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Theorem utopsnneip 21341
Description: The neighborhoods of a point  P for the topology induced by an uniform space  U. (Contributed by Thierry Arnoux, 13-Jan-2018.)
Hypothesis
Ref Expression
utoptop.1  |-  J  =  (unifTop `  U )
Assertion
Ref Expression
utopsnneip  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  J
) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
Distinct variable groups:    v, P    v, U    v, X
Allowed substitution hint:    J( v)

Proof of Theorem utopsnneip
Dummy variables  p  a  b  q  r are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 utoptop.1 . 2  |-  J  =  (unifTop `  U )
2 fveq2 5879 . . . . . 6  |-  ( r  =  p  ->  (
( q  e.  X  |->  ran  ( v  e.  U  |->  ( v " { q } ) ) ) `  r
)  =  ( ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v " {
q } ) ) ) `  p ) )
32eleq2d 2534 . . . . 5  |-  ( r  =  p  ->  (
b  e.  ( ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v " {
q } ) ) ) `  r )  <-> 
b  e.  ( ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v " {
q } ) ) ) `  p ) ) )
43cbvralv 3005 . . . 4  |-  ( A. r  e.  b  b  e.  ( ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v
" { q } ) ) ) `  r )  <->  A. p  e.  b  b  e.  ( ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v
" { q } ) ) ) `  p ) )
5 eleq1 2537 . . . . 5  |-  ( b  =  a  ->  (
b  e.  ( ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v " {
q } ) ) ) `  p )  <-> 
a  e.  ( ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v " {
q } ) ) ) `  p ) ) )
65raleqbi1dv 2981 . . . 4  |-  ( b  =  a  ->  ( A. p  e.  b 
b  e.  ( ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v " {
q } ) ) ) `  p )  <->  A. p  e.  a 
a  e.  ( ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v " {
q } ) ) ) `  p ) ) )
74, 6syl5bb 265 . . 3  |-  ( b  =  a  ->  ( A. r  e.  b 
b  e.  ( ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v " {
q } ) ) ) `  r )  <->  A. p  e.  a 
a  e.  ( ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v " {
q } ) ) ) `  p ) ) )
87cbvrabv 3030 . 2  |-  { b  e.  ~P X  |  A. r  e.  b 
b  e.  ( ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v " {
q } ) ) ) `  r ) }  =  { a  e.  ~P X  |  A. p  e.  a 
a  e.  ( ( q  e.  X  |->  ran  ( v  e.  U  |->  ( v " {
q } ) ) ) `  p ) }
9 simpl 464 . . . . . . 7  |-  ( ( q  =  p  /\  v  e.  U )  ->  q  =  p )
109sneqd 3971 . . . . . 6  |-  ( ( q  =  p  /\  v  e.  U )  ->  { q }  =  { p } )
1110imaeq2d 5174 . . . . 5  |-  ( ( q  =  p  /\  v  e.  U )  ->  ( v " {
q } )  =  ( v " {
p } ) )
1211mpteq2dva 4482 . . . 4  |-  ( q  =  p  ->  (
v  e.  U  |->  ( v " { q } ) )  =  ( v  e.  U  |->  ( v " {
p } ) ) )
1312rneqd 5068 . . 3  |-  ( q  =  p  ->  ran  ( v  e.  U  |->  ( v " {
q } ) )  =  ran  ( v  e.  U  |->  ( v
" { p }
) ) )
1413cbvmptv 4488 . 2  |-  ( q  e.  X  |->  ran  (
v  e.  U  |->  ( v " { q } ) ) )  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v
" { p }
) ) )
151, 8, 14utopsnneiplem 21340 1  |-  ( ( U  e.  (UnifOn `  X )  /\  P  e.  X )  ->  (
( nei `  J
) `  { P } )  =  ran  ( v  e.  U  |->  ( v " { P } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 376    = wceq 1452    e. wcel 1904   A.wral 2756   {crab 2760   ~Pcpw 3942   {csn 3959    |-> cmpt 4454   ran crn 4840   "cima 4842   ` cfv 5589   neicnei 20190  UnifOncust 21292  unifTopcutop 21323
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-rep 4508  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-reu 2763  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-tp 3964  df-op 3966  df-uni 4191  df-int 4227  df-iun 4271  df-br 4396  df-opab 4455  df-mpt 4456  df-tr 4491  df-eprel 4750  df-id 4754  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-ord 5433  df-on 5434  df-lim 5435  df-suc 5436  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-om 6712  df-wrecs 7046  df-recs 7108  df-rdg 7146  df-1o 7200  df-oadd 7204  df-er 7381  df-en 7588  df-fin 7591  df-fi 7943  df-top 19998  df-nei 20191  df-ust 21293  df-utop 21324
This theorem is referenced by:  utopsnnei  21342  utopreg  21345  neipcfilu  21389
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