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Theorem neiptopreu 20142
Description: If, to each element  P of a set  X, we associate a set  ( N `  P ) fulfilling the properties Vi, Vii, Viii and property Viv of [BourbakiTop1] p. I.2. , corresponding to ssnei 20119, innei 20134, elnei 20120 and neissex 20136, then there is a unique topology  j such that for any point  p,  ( N `  p ) is the set of neighborhoods of  p. Proposition 2 of [BourbakiTop1] p. I.3. This can be used to build a topology from a set of neighborhoods. Note that the additional condition that  X is a neighborhood of all points was added. (Contributed by Thierry Arnoux, 6-Jan-2018.)
Hypotheses
Ref Expression
neiptop.o  |-  J  =  { a  e.  ~P X  |  A. p  e.  a  a  e.  ( N `  p ) }
neiptop.0  |-  ( ph  ->  N : X --> ~P ~P X )
neiptop.1  |-  ( ( ( ( ph  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p ) )  -> 
b  e.  ( N `
 p ) )
neiptop.2  |-  ( (
ph  /\  p  e.  X )  ->  ( fi `  ( N `  p ) )  C_  ( N `  p ) )
neiptop.3  |-  ( ( ( ph  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  p  e.  a )
neiptop.4  |-  ( ( ( ph  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. b  e.  ( N `  p
) A. q  e.  b  a  e.  ( N `  q ) )
neiptop.5  |-  ( (
ph  /\  p  e.  X )  ->  X  e.  ( N `  p
) )
Assertion
Ref Expression
neiptopreu  |-  ( ph  ->  E! j  e.  (TopOn `  X ) N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )
Distinct variable groups:    p, a, N    X, a, b, p    J, a, p    X, p    ph, p    N, b    X, b    ph, a, b, q, p    N, p, q    X, q    ph, q    j, a, b, J, p    j,
q, N    j, X    ph, j
Allowed substitution hint:    J( q)

Proof of Theorem neiptopreu
StepHypRef Expression
1 neiptop.o . . . . 5  |-  J  =  { a  e.  ~P X  |  A. p  e.  a  a  e.  ( N `  p ) }
2 neiptop.0 . . . . 5  |-  ( ph  ->  N : X --> ~P ~P X )
3 neiptop.1 . . . . 5  |-  ( ( ( ( ph  /\  p  e.  X )  /\  a  C_  b  /\  b  C_  X )  /\  a  e.  ( N `  p ) )  -> 
b  e.  ( N `
 p ) )
4 neiptop.2 . . . . 5  |-  ( (
ph  /\  p  e.  X )  ->  ( fi `  ( N `  p ) )  C_  ( N `  p ) )
5 neiptop.3 . . . . 5  |-  ( ( ( ph  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  p  e.  a )
6 neiptop.4 . . . . 5  |-  ( ( ( ph  /\  p  e.  X )  /\  a  e.  ( N `  p
) )  ->  E. b  e.  ( N `  p
) A. q  e.  b  a  e.  ( N `  q ) )
7 neiptop.5 . . . . 5  |-  ( (
ph  /\  p  e.  X )  ->  X  e.  ( N `  p
) )
81, 2, 3, 4, 5, 6, 7neiptoptop 20140 . . . 4  |-  ( ph  ->  J  e.  Top )
9 eqid 2450 . . . . 5  |-  U. J  =  U. J
109toptopon 19941 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
118, 10sylib 200 . . 3  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
121, 2, 3, 4, 5, 6, 7neiptopuni 20139 . . . 4  |-  ( ph  ->  X  =  U. J
)
1312fveq2d 5867 . . 3  |-  ( ph  ->  (TopOn `  X )  =  (TopOn `  U. J ) )
1411, 13eleqtrrd 2531 . 2  |-  ( ph  ->  J  e.  (TopOn `  X ) )
151, 2, 3, 4, 5, 6, 7neiptopnei 20141 . 2  |-  ( ph  ->  N  =  ( p  e.  X  |->  ( ( nei `  J ) `
 { p }
) ) )
16 nfv 1760 . . . . . . . . . 10  |-  F/ p
( ph  /\  j  e.  (TopOn `  X )
)
17 nfmpt1 4491 . . . . . . . . . . 11  |-  F/_ p
( p  e.  X  |->  ( ( nei `  j
) `  { p } ) )
1817nfeq2 2606 . . . . . . . . . 10  |-  F/ p  N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )
1916, 18nfan 2010 . . . . . . . . 9  |-  F/ p
( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )
20 nfv 1760 . . . . . . . . 9  |-  F/ p  b  C_  X
2119, 20nfan 2010 . . . . . . . 8  |-  F/ p
( ( ( ph  /\  j  e.  (TopOn `  X ) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) ) )  /\  b  C_  X )
22 simpllr 768 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  j  e.  (TopOn `  X ) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) ) )  /\  b  C_  X )  /\  p  e.  b )  ->  N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) ) )
23 simpr 463 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )  /\  b  C_  X )  ->  b  C_  X )
2423sselda 3431 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  j  e.  (TopOn `  X ) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) ) )  /\  b  C_  X )  /\  p  e.  b )  ->  p  e.  X )
25 id 22 . . . . . . . . . . . 12  |-  ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) )  ->  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )
26 fvex 5873 . . . . . . . . . . . . 13  |-  ( ( nei `  j ) `
 { p }
)  e.  _V
2726a1i 11 . . . . . . . . . . . 12  |-  ( ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )  /\  p  e.  X )  ->  (
( nei `  j
) `  { p } )  e.  _V )
2825, 27fvmpt2d 5957 . . . . . . . . . . 11  |-  ( ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )  /\  p  e.  X )  ->  ( N `  p )  =  ( ( nei `  j ) `  {
p } ) )
2922, 24, 28syl2anc 666 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  j  e.  (TopOn `  X ) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) ) )  /\  b  C_  X )  /\  p  e.  b )  ->  ( N `  p
)  =  ( ( nei `  j ) `
 { p }
) )
3029eqcomd 2456 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  j  e.  (TopOn `  X ) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) ) )  /\  b  C_  X )  /\  p  e.  b )  ->  ( ( nei `  j
) `  { p } )  =  ( N `  p ) )
3130eleq2d 2513 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  j  e.  (TopOn `  X ) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) ) )  /\  b  C_  X )  /\  p  e.  b )  ->  ( b  e.  ( ( nei `  j
) `  { p } )  <->  b  e.  ( N `  p ) ) )
3221, 31ralbida 2820 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )  /\  b  C_  X )  ->  ( A. p  e.  b 
b  e.  ( ( nei `  j ) `
 { p }
)  <->  A. p  e.  b  b  e.  ( N `
 p ) ) )
3332pm5.32da 646 . . . . . 6  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  ( ( b 
C_  X  /\  A. p  e.  b  b  e.  ( ( nei `  j
) `  { p } ) )  <->  ( b  C_  X  /\  A. p  e.  b  b  e.  ( N `  p ) ) ) )
34 simpllr 768 . . . . . . . . 9  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )  /\  b  e.  j )  ->  j  e.  (TopOn `  X )
)
35 simpr 463 . . . . . . . . 9  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )  /\  b  e.  j )  ->  b  e.  j )
36 toponss 19937 . . . . . . . . 9  |-  ( ( j  e.  (TopOn `  X )  /\  b  e.  j )  ->  b  C_  X )
3734, 35, 36syl2anc 666 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )  /\  b  e.  j )  ->  b  C_  X )
38 topontop 19934 . . . . . . . . . . 11  |-  ( j  e.  (TopOn `  X
)  ->  j  e.  Top )
3938ad2antlr 732 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  j  e.  Top )
40 opnnei 20129 . . . . . . . . . 10  |-  ( j  e.  Top  ->  (
b  e.  j  <->  A. p  e.  b  b  e.  ( ( nei `  j
) `  { p } ) ) )
4139, 40syl 17 . . . . . . . . 9  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  ( b  e.  j  <->  A. p  e.  b  b  e.  ( ( nei `  j ) `
 { p }
) ) )
4241biimpa 487 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )  /\  b  e.  j )  ->  A. p  e.  b  b  e.  ( ( nei `  j
) `  { p } ) )
4337, 42jca 535 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )  /\  b  e.  j )  ->  (
b  C_  X  /\  A. p  e.  b  b  e.  ( ( nei `  j ) `  {
p } ) ) )
4441biimpar 488 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )  /\  A. p  e.  b  b  e.  ( ( nei `  j
) `  { p } ) )  -> 
b  e.  j )
4544adantrl 721 . . . . . . 7  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )  /\  ( b 
C_  X  /\  A. p  e.  b  b  e.  ( ( nei `  j
) `  { p } ) ) )  ->  b  e.  j )
4643, 45impbida 842 . . . . . 6  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  ( b  e.  j  <->  ( b  C_  X  /\  A. p  e.  b  b  e.  ( ( nei `  j
) `  { p } ) ) ) )
471neipeltop 20138 . . . . . . 7  |-  ( b  e.  J  <->  ( b  C_  X  /\  A. p  e.  b  b  e.  ( N `  p ) ) )
4847a1i 11 . . . . . 6  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  ( b  e.  J  <->  ( b  C_  X  /\  A. p  e.  b  b  e.  ( N `  p ) ) ) )
4933, 46, 483bitr4d 289 . . . . 5  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  ( b  e.  j  <->  b  e.  J
) )
5049eqrdv 2448 . . . 4  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  j  =  J )
5150ex 436 . . 3  |-  ( (
ph  /\  j  e.  (TopOn `  X ) )  ->  ( N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) )  -> 
j  =  J ) )
5251ralrimiva 2801 . 2  |-  ( ph  ->  A. j  e.  (TopOn `  X ) ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) )  ->  j  =  J ) )
53 simpl 459 . . . . . . 7  |-  ( ( j  =  J  /\  p  e.  X )  ->  j  =  J )
5453fveq2d 5867 . . . . . 6  |-  ( ( j  =  J  /\  p  e.  X )  ->  ( nei `  j
)  =  ( nei `  J ) )
5554fveq1d 5865 . . . . 5  |-  ( ( j  =  J  /\  p  e.  X )  ->  ( ( nei `  j
) `  { p } )  =  ( ( nei `  J
) `  { p } ) )
5655mpteq2dva 4488 . . . 4  |-  ( j  =  J  ->  (
p  e.  X  |->  ( ( nei `  j
) `  { p } ) )  =  ( p  e.  X  |->  ( ( nei `  J
) `  { p } ) ) )
5756eqeq2d 2460 . . 3  |-  ( j  =  J  ->  ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )  <->  N  =  ( p  e.  X  |->  ( ( nei `  J
) `  { p } ) ) ) )
5857eqreu 3229 . 2  |-  ( ( J  e.  (TopOn `  X )  /\  N  =  ( p  e.  X  |->  ( ( nei `  J ) `  {
p } ) )  /\  A. j  e.  (TopOn `  X )
( N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) )  -> 
j  =  J ) )  ->  E! j  e.  (TopOn `  X ) N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) ) )
5914, 15, 52, 58syl3anc 1267 1  |-  ( ph  ->  E! j  e.  (TopOn `  X ) N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 188    /\ wa 371    /\ w3a 984    = wceq 1443    e. wcel 1886   A.wral 2736   E.wrex 2737   E!wreu 2738   {crab 2740   _Vcvv 3044    C_ wss 3403   ~Pcpw 3950   {csn 3967   U.cuni 4197    |-> cmpt 4460   -->wf 5577   ` cfv 5581   ficfi 7921   Topctop 19910  TopOnctopon 19911   neicnei 20106
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-rep 4514  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-csb 3363  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-iun 4279  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-pred 5379  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-ov 6291  df-oprab 6292  df-mpt2 6293  df-om 6690  df-wrecs 7025  df-recs 7087  df-rdg 7125  df-1o 7179  df-oadd 7183  df-er 7360  df-en 7567  df-fin 7570  df-fi 7922  df-top 19914  df-topon 19916  df-ntr 20028  df-nei 20107
This theorem is referenced by:  ustuqtop  21254
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