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Theorem ustuqtop 20875
Description: For a given uniform structure  U on a set  X, there is a unique topology  j such that the set  ran  ( v  e.  U  |->  ( v
" { p }
) ) is the filter of the neighborhoods of  p for that topology. Proposition 1 of [BourbakiTop1] p. II.3. (Contributed by Thierry Arnoux, 11-Jan-2018.)
Hypothesis
Ref Expression
utopustuq.1  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
Assertion
Ref Expression
ustuqtop  |-  ( U  e.  (UnifOn `  X
)  ->  E! j  e.  (TopOn `  X ) A. p  e.  X  ( N `  p )  =  ( ( nei `  j ) `  {
p } ) )
Distinct variable groups:    v, p, U    X, p, v, j   
j, N, p    v,
j, U    j, X
Allowed substitution hint:    N( v)

Proof of Theorem ustuqtop
Dummy variables  a 
b  c  r  x are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5872 . . . . . . 7  |-  ( p  =  r  ->  ( N `  p )  =  ( N `  r ) )
21eleq2d 2527 . . . . . 6  |-  ( p  =  r  ->  (
c  e.  ( N `
 p )  <->  c  e.  ( N `  r ) ) )
32cbvralv 3084 . . . . 5  |-  ( A. p  e.  c  c  e.  ( N `  p
)  <->  A. r  e.  c  c  e.  ( N `
 r ) )
4 eleq1 2529 . . . . . 6  |-  ( c  =  a  ->  (
c  e.  ( N `
 p )  <->  a  e.  ( N `  p ) ) )
54raleqbi1dv 3062 . . . . 5  |-  ( c  =  a  ->  ( A. p  e.  c 
c  e.  ( N `
 p )  <->  A. p  e.  a  a  e.  ( N `  p ) ) )
63, 5syl5bbr 259 . . . 4  |-  ( c  =  a  ->  ( A. r  e.  c 
c  e.  ( N `
 r )  <->  A. p  e.  a  a  e.  ( N `  p ) ) )
76cbvrabv 3108 . . 3  |-  { c  e.  ~P X  |  A. r  e.  c 
c  e.  ( N `
 r ) }  =  { a  e. 
~P X  |  A. p  e.  a  a  e.  ( N `  p
) }
8 utopustuq.1 . . . 4  |-  N  =  ( p  e.  X  |->  ran  ( v  e.  U  |->  ( v " { p } ) ) )
98ustuqtop0 20869 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  N : X
--> ~P ~P X )
108ustuqtop1 20870 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  a  e.  ( N `  p ) )  ->  b  e.  ( N `  p ) )
118ustuqtop2 20871 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( fi `  ( N `  p ) )  C_  ( N `  p ) )
128ustuqtop3 20872 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  p  e.  a )
138ustuqtop4 20873 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. b  e.  ( N `  p
) A. x  e.  b  a  e.  ( N `  x ) )
148ustuqtop5 20874 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  e.  ( N `  p
) )
157, 9, 10, 11, 12, 13, 14neiptopreu 19761 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  E! j  e.  (TopOn `  X ) N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) ) )
169feqmptd 5926 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  N  =  ( p  e.  X  |->  ( N `  p
) ) )
1716eqeq1d 2459 . . . 4  |-  ( U  e.  (UnifOn `  X
)  ->  ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) )  <-> 
( p  e.  X  |->  ( N `  p
) )  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) ) )
18 fvex 5882 . . . . . 6  |-  ( N `
 p )  e. 
_V
1918rgenw 2818 . . . . 5  |-  A. p  e.  X  ( N `  p )  e.  _V
20 mpteqb 5971 . . . . 5  |-  ( A. p  e.  X  ( N `  p )  e.  _V  ->  ( (
p  e.  X  |->  ( N `  p ) )  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )  <->  A. p  e.  X  ( N `  p )  =  ( ( nei `  j
) `  { p } ) ) )
2119, 20ax-mp 5 . . . 4  |-  ( ( p  e.  X  |->  ( N `  p ) )  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )  <->  A. p  e.  X  ( N `  p )  =  ( ( nei `  j
) `  { p } ) )
2217, 21syl6bb 261 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) )  <->  A. p  e.  X  ( N `  p )  =  ( ( nei `  j ) `  {
p } ) ) )
2322reubidv 3042 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  ( E! j  e.  (TopOn `  X
) N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) )  <->  E! j  e.  (TopOn `  X ) A. p  e.  X  ( N `  p )  =  ( ( nei `  j ) `  {
p } ) ) )
2415, 23mpbid 210 1  |-  ( U  e.  (UnifOn `  X
)  ->  E! j  e.  (TopOn `  X ) A. p  e.  X  ( N `  p )  =  ( ( nei `  j ) `  {
p } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    = wceq 1395    e. wcel 1819   A.wral 2807   E!wreu 2809   {crab 2811   _Vcvv 3109   ~Pcpw 4015   {csn 4032    |-> cmpt 4515   ran crn 5009   "cima 5011   ` cfv 5594  TopOnctopon 19522   neicnei 19725  UnifOncust 20828
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-rep 4568  ax-sep 4578  ax-nul 4586  ax-pow 4634  ax-pr 4695  ax-un 6591
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3111  df-sbc 3328  df-csb 3431  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-pss 3487  df-nul 3794  df-if 3945  df-pw 4017  df-sn 4033  df-pr 4035  df-tp 4037  df-op 4039  df-uni 4252  df-int 4289  df-iun 4334  df-br 4457  df-opab 4516  df-mpt 4517  df-tr 4551  df-eprel 4800  df-id 4804  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891  df-lim 4892  df-suc 4893  df-xp 5014  df-rel 5015  df-cnv 5016  df-co 5017  df-dm 5018  df-rn 5019  df-res 5020  df-ima 5021  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 6299  df-oprab 6300  df-mpt2 6301  df-om 6700  df-recs 7060  df-rdg 7094  df-1o 7148  df-oadd 7152  df-er 7329  df-en 7536  df-fin 7539  df-fi 7889  df-top 19526  df-topon 19529  df-ntr 19648  df-nei 19726  df-ust 20829
This theorem is referenced by: (None)
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