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Theorem ustuqtop 20481
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 5864 . . . . . . 7  |-  ( p  =  r  ->  ( N `  p )  =  ( N `  r ) )
21eleq2d 2537 . . . . . 6  |-  ( p  =  r  ->  (
c  e.  ( N `
 p )  <->  c  e.  ( N `  r ) ) )
32cbvralv 3088 . . . . 5  |-  ( A. p  e.  c  c  e.  ( N `  p
)  <->  A. r  e.  c  c  e.  ( N `
 r ) )
4 eleq1 2539 . . . . . 6  |-  ( c  =  a  ->  (
c  e.  ( N `
 p )  <->  a  e.  ( N `  p ) ) )
54raleqbi1dv 3066 . . . . 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 3112 . . 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 20475 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  N : X
--> ~P ~P X )
108ustuqtop1 20476 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  a  e.  ( N `  p ) )  ->  b  e.  ( N `  p ) )
118ustuqtop2 20477 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( fi `  ( N `  p ) )  C_  ( N `  p ) )
128ustuqtop3 20478 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  p  e.  a )
138ustuqtop4 20479 . . 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 20480 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  e.  ( N `  p
) )
157, 9, 10, 11, 12, 13, 14neiptopreu 19397 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  E! j  e.  (TopOn `  X ) N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) ) )
169feqmptd 5918 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  N  =  ( p  e.  X  |->  ( N `  p
) ) )
1716eqeq1d 2469 . . . 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 5874 . . . . . 6  |-  ( N `
 p )  e. 
_V
1918rgenw 2825 . . . . 5  |-  A. p  e.  X  ( N `  p )  e.  _V
20 mpteqb 5962 . . . . 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 3046 . 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 1379    e. wcel 1767   A.wral 2814   E!wreu 2816   {crab 2818   _Vcvv 3113   ~Pcpw 4010   {csn 4027    |-> cmpt 4505   ran crn 5000   "cima 5002   ` cfv 5586  TopOnctopon 19159   neicnei 19361  UnifOncust 20434
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 4558  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
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 2819  df-rex 2820  df-reu 2821  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-int 4283  df-iun 4327  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-om 6679  df-recs 7039  df-rdg 7073  df-1o 7127  df-oadd 7131  df-er 7308  df-en 7514  df-fin 7517  df-fi 7867  df-top 19163  df-topon 19166  df-ntr 19284  df-nei 19362  df-ust 20435
This theorem is referenced by: (None)
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