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Theorem neiptopreu 19500
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 19477, innei 19492, elnei 19478 and neissex 19494, 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 19498 . . . 4  |-  ( ph  ->  J  e.  Top )
9 eqid 2441 . . . . 5  |-  U. J  =  U. J
109toptopon 19301 . . . 4  |-  ( J  e.  Top  <->  J  e.  (TopOn `  U. J ) )
118, 10sylib 196 . . 3  |-  ( ph  ->  J  e.  (TopOn `  U. J ) )
121, 2, 3, 4, 5, 6, 7neiptopuni 19497 . . . 4  |-  ( ph  ->  X  =  U. J
)
1312fveq2d 5856 . . 3  |-  ( ph  ->  (TopOn `  X )  =  (TopOn `  U. J ) )
1411, 13eleqtrrd 2532 . 2  |-  ( ph  ->  J  e.  (TopOn `  X ) )
151, 2, 3, 4, 5, 6, 7neiptopnei 19499 . 2  |-  ( ph  ->  N  =  ( p  e.  X  |->  ( ( nei `  J ) `
 { p }
) ) )
16 nfv 1692 . . . . . . . . . 10  |-  F/ p
( ph  /\  j  e.  (TopOn `  X )
)
17 nfmpt1 4522 . . . . . . . . . . 11  |-  F/_ p
( p  e.  X  |->  ( ( nei `  j
) `  { p } ) )
1817nfeq2 2620 . . . . . . . . . 10  |-  F/ p  N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )
1916, 18nfan 1912 . . . . . . . . 9  |-  F/ p
( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )
20 nfv 1692 . . . . . . . . 9  |-  F/ p  b  C_  X
2119, 20nfan 1912 . . . . . . . 8  |-  F/ p
( ( ( ph  /\  j  e.  (TopOn `  X ) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) ) )  /\  b  C_  X )
22 simpllr 758 . . . . . . . . . . 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 461 . . . . . . . . . . . 12  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )  /\  b  C_  X )  ->  b  C_  X )
2423sselda 3486 . . . . . . . . . . 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 5862 . . . . . . . . . . . . 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 5946 . . . . . . . . . . 11  |-  ( ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )  /\  p  e.  X )  ->  ( N `  p )  =  ( ( nei `  j ) `  {
p } ) )
2922, 24, 28syl2anc 661 . . . . . . . . . 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 2449 . . . . . . . . 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 2511 . . . . . . . 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 2874 . . . . . . 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 641 . . . . . 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 758 . . . . . . . . 9  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )  /\  b  e.  j )  ->  j  e.  (TopOn `  X )
)
35 simpr 461 . . . . . . . . 9  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )  /\  b  e.  j )  ->  b  e.  j )
36 toponss 19297 . . . . . . . . 9  |-  ( ( j  e.  (TopOn `  X )  /\  b  e.  j )  ->  b  C_  X )
3734, 35, 36syl2anc 661 . . . . . . . 8  |-  ( ( ( ( ph  /\  j  e.  (TopOn `  X
) )  /\  N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) ) )  /\  b  e.  j )  ->  b  C_  X )
38 topontop 19294 . . . . . . . . . . 11  |-  ( j  e.  (TopOn `  X
)  ->  j  e.  Top )
3938ad2antlr 726 . . . . . . . . . 10  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  j  e.  Top )
40 opnnei 19487 . . . . . . . . . 10  |-  ( j  e.  Top  ->  (
b  e.  j  <->  A. p  e.  b  b  e.  ( ( nei `  j
) `  { p } ) ) )
4139, 40syl 16 . . . . . . . . 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 484 . . . . . . . 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 532 . . . . . . 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 485 . . . . . . . 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 715 . . . . . . 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 830 . . . . . 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 19496 . . . . . . 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 285 . . . . 5  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  ( b  e.  j  <->  b  e.  J
) )
5049eqrdv 2438 . . . 4  |-  ( ( ( ph  /\  j  e.  (TopOn `  X )
)  /\  N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) ) )  ->  j  =  J )
5150ex 434 . . 3  |-  ( (
ph  /\  j  e.  (TopOn `  X ) )  ->  ( N  =  ( p  e.  X  |->  ( ( nei `  j
) `  { p } ) )  -> 
j  =  J ) )
5251ralrimiva 2855 . 2  |-  ( ph  ->  A. j  e.  (TopOn `  X ) ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `  {
p } ) )  ->  j  =  J ) )
53 simpl 457 . . . . . . 7  |-  ( ( j  =  J  /\  p  e.  X )  ->  j  =  J )
5453fveq2d 5856 . . . . . 6  |-  ( ( j  =  J  /\  p  e.  X )  ->  ( nei `  j
)  =  ( nei `  J ) )
5554fveq1d 5854 . . . . 5  |-  ( ( j  =  J  /\  p  e.  X )  ->  ( ( nei `  j
) `  { p } )  =  ( ( nei `  J
) `  { p } ) )
5655mpteq2dva 4519 . . . 4  |-  ( j  =  J  ->  (
p  e.  X  |->  ( ( nei `  j
) `  { p } ) )  =  ( p  e.  X  |->  ( ( nei `  J
) `  { p } ) ) )
5756eqeq2d 2455 . . 3  |-  ( j  =  J  ->  ( N  =  ( p  e.  X  |->  ( ( nei `  j ) `
 { p }
) )  <->  N  =  ( p  e.  X  |->  ( ( nei `  J
) `  { p } ) ) ) )
5857eqreu 3275 . 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 1227 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 184    /\ wa 369    /\ w3a 972    = wceq 1381    e. wcel 1802   A.wral 2791   E.wrex 2792   E!wreu 2793   {crab 2795   _Vcvv 3093    C_ wss 3458   ~Pcpw 3993   {csn 4010   U.cuni 4230    |-> cmpt 4491   -->wf 5570   ` cfv 5574   ficfi 7868   Topctop 19261  TopOnctopon 19262   neicnei 19464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-rep 4544  ax-sep 4554  ax-nul 4562  ax-pow 4611  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 973  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-reu 2798  df-rab 2800  df-v 3095  df-sbc 3312  df-csb 3418  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-pss 3474  df-nul 3768  df-if 3923  df-pw 3995  df-sn 4011  df-pr 4013  df-tp 4015  df-op 4017  df-uni 4231  df-int 4268  df-iun 4313  df-br 4434  df-opab 4492  df-mpt 4493  df-tr 4527  df-eprel 4777  df-id 4781  df-po 4786  df-so 4787  df-fr 4824  df-we 4826  df-ord 4867  df-on 4868  df-lim 4869  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-res 4997  df-ima 4998  df-iota 5537  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-fv 5582  df-ov 6280  df-oprab 6281  df-mpt2 6282  df-om 6682  df-recs 7040  df-rdg 7074  df-1o 7128  df-oadd 7132  df-er 7309  df-en 7515  df-fin 7518  df-fi 7869  df-top 19266  df-topon 19269  df-ntr 19387  df-nei 19465
This theorem is referenced by:  ustuqtop  20615
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