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Theorem ustuqtop 19826
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 5696 . . . . . . 7  |-  ( p  =  r  ->  ( N `  p )  =  ( N `  r ) )
21eleq2d 2510 . . . . . 6  |-  ( p  =  r  ->  (
c  e.  ( N `
 p )  <->  c  e.  ( N `  r ) ) )
32cbvralv 2952 . . . . 5  |-  ( A. p  e.  c  c  e.  ( N `  p
)  <->  A. r  e.  c  c  e.  ( N `
 r ) )
4 eleq1 2503 . . . . . 6  |-  ( c  =  a  ->  (
c  e.  ( N `
 p )  <->  a  e.  ( N `  p ) ) )
54raleqbi1dv 2930 . . . . 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 2976 . . 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 19820 . . 3  |-  ( U  e.  (UnifOn `  X
)  ->  N : X
--> ~P ~P X )
108ustuqtop1 19821 . . 3  |-  ( ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X
)  /\  a  C_  b  /\  b  C_  X
)  /\  a  e.  ( N `  p ) )  ->  b  e.  ( N `  p ) )
118ustuqtop2 19822 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  ( fi `  ( N `  p ) )  C_  ( N `  p ) )
128ustuqtop3 19823 . . 3  |-  ( ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  p  e.  a )
138ustuqtop4 19824 . . 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 19825 . . 3  |-  ( ( U  e.  (UnifOn `  X )  /\  p  e.  X )  ->  X  e.  ( N `  p
) )
157, 9, 10, 11, 12, 13, 14neiptopreu 18742 . 2  |-  ( U  e.  (UnifOn `  X
)  ->  E! j  e.  (TopOn `  X ) N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) ) )
169feqmptd 5749 . . . . 5  |-  ( U  e.  (UnifOn `  X
)  ->  N  =  ( p  e.  X  |->  ( N `  p
) ) )
1716eqeq1d 2451 . . . 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 5706 . . . . . 6  |-  ( N `
 p )  e. 
_V
1918rgenw 2788 . . . . 5  |-  A. p  e.  X  ( N `  p )  e.  _V
20 mpteqb 5793 . . . . 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 2910 . 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 1369    e. wcel 1756   A.wral 2720   E!wreu 2722   {crab 2724   _Vcvv 2977   ~Pcpw 3865   {csn 3882    e. cmpt 4355   ran crn 4846   "cima 4848   ` cfv 5423  TopOnctopon 18504   neicnei 18706  UnifOncust 19779
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 4408  ax-sep 4418  ax-nul 4426  ax-pow 4475  ax-pr 4536  ax-un 6377
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 2573  df-ne 2613  df-ral 2725  df-rex 2726  df-reu 2727  df-rab 2729  df-v 2979  df-sbc 3192  df-csb 3294  df-dif 3336  df-un 3338  df-in 3340  df-ss 3347  df-pss 3349  df-nul 3643  df-if 3797  df-pw 3867  df-sn 3883  df-pr 3885  df-tp 3887  df-op 3889  df-uni 4097  df-int 4134  df-iun 4178  df-br 4298  df-opab 4356  df-mpt 4357  df-tr 4391  df-eprel 4637  df-id 4641  df-po 4646  df-so 4647  df-fr 4684  df-we 4686  df-ord 4727  df-on 4728  df-lim 4729  df-suc 4730  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5386  df-fun 5425  df-fn 5426  df-f 5427  df-f1 5428  df-fo 5429  df-f1o 5430  df-fv 5431  df-ov 6099  df-oprab 6100  df-mpt2 6101  df-om 6482  df-recs 6837  df-rdg 6871  df-1o 6925  df-oadd 6929  df-er 7106  df-en 7316  df-fin 7319  df-fi 7666  df-top 18508  df-topon 18511  df-ntr 18629  df-nei 18707  df-ust 19780
This theorem is referenced by: (None)
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